Friday, December 11, 2009

Math 8 Unit Plan - Shape & Space & Measurement

Name: Rory Whitbread
Title of unit and grade / course: Shape & Space & Their Measurement / Mathematics 8

1) Rationale and connections:
a) It is important for students to learn the topic of space, shape and the measurement of each as they are broad reaching and provide a base for many future topics in mathematics and other subjects and even general everyday life. This subject area is also an excellent opportunity to include history with mathematics in the discussion of how varying shapes and measurements were used by other civilizations in other times. These subjects are included in the IRPs as many sections of higher grade mathematics depend deeply on their understanding.

b) Historical origins of this topic are vast. Many civilizations had broad working knowledge of geometrical shapes and their measurement. These were used for the buying and selling of commodities and for the division and taxation of land. Of course, mathematics was also performed by academics and for recreation and many of the ancient findings produced are still used widely today. The Pythagorean Theorem is one of the most famous in all of mathematics and is named for Pythagoras and his followers, but it seems it was being used by many more civilizations than just the early Greeks.

c) Shape, Space and Measurement relate closely to life outside of mathematics. First, they provide for a base of good spatial thought and reasoning. Second, they are topics that are easily found in every day life; from measuring the dimensions of a box that needs to made, to knowing how tall a ladder will be that will be able to reach the roof, to the age old measurement of land and structures. Having a good understand of these topics also helps with estimation - a valuable skill in a fast paced world.

2) Balanced teaching, assessment and evaluation plan
a) This is a very important unit of the Mathematics 8 curriculum and will therefore be worth 10% of the weight for grading. For this reason, multiple forms of assessment will also be used, with teacher assessment for a quiz and unit test and peer and student assessment utilized for a group project that will be completed throughout the 10 classes. The midway quiz and unit test will comprise 1% and 4% respectively. The remaining 5% will be assessed from the group project which will have 2.5% resulting from self assessment through journal work and 2.5% as evaluated by the group. With these forms of assessment in mind, there will be a good balance between summative and formative assessment as well as between instrumental and relational learning. All things considered, it is hoped that this will one of the favourite units of the students’ Math 8 year.

b) House Building Blueprints Project – The class project that will be worked on throughout this unit will be the basic design of a house or structure from the past, present or future. This project will be completed in groups of three or four with the group presenting their 2 or 3 dimensional final product to the class.


c)
Lesson topic Teaching strategies / approaches used
1) Surface Area Area and Surface Area will be reviewed and discussed. Specific applications for right rectangular and right triangular prisms and for right cylinders will be explored. The class will be placed into groups of three or four that will also be used for the unit project. Within these groups, tactile analysis of sample shapes will be used and nets will be constructed.

2) Surface Area Continued Students will be introduced to the volumes of the shapes explored last day. Relations of Surface Area to the dimensions and Perimeters of shapes will be explored. Students will then complete worksheets with questions relating to surface area.

3) Group Project Work Students will be introduced to the Unit Group Project (details given at end of unit plan). They will be given the full class to come up with a plan for what type of building they will create and which shapes will be included and how their projects will be presented to the class.

4) Pythagorean Theorem Students will be introduced to the Pythagorean Theorem with approximately 15 minutes of lecture involving concept and history. Then students will be split up into their groups to explore triangular shapes and perfect Pythagorean Triangles. They will be given knotted strings to explore these the way people would have thousands of years ago. Human triangles will be performed to conclude the class.

5) Group Work / Quiz Students will use the first half of the class to work on their group projects, with specific emphasis on the right triangular portions of their structures. The remaining half of the class will be used for a quiz on Surface Area and the Pythagorean Theorem.

6) Volumes of Shapes Early class will be devoted to lecture and discussion of the volumes of the shapes from lesson one. Relationships between Surface Area and Volume will be explored. Students will spend the last portion of class in groups discussing their group projects.

7) Group Project Work Students will have the entire class to work on their group projects. They will be asked to consider and discuss the cost of heating these structures as an application of volume.

8) Review Work and Class Presentations The first portion of the class will be devoted to individual work on reviewing the material in the unit thus far. The second portion will allow students to complete final work on their group projects and get ready for presentations. The third portion will be given for the initial presentations.

9) Group Presentations All remaining groups will present to the class and concepts will be reviewed as appropriate throughout.

10) Unit Test The early class time will be devoted to a short review and class questions. The remainder of class time will be devoted to a Unit Test which will evaluate what was learned over the previous nine classes and will constitute 4% of the final grade.


3) In detail:

Surface Area

Rationale: To give students an understanding of surface area and its applications. The Surface Areas of various shapes will be explored in a hands-on fashion.

Instructional Objectives: Each student will understand the concept of Surface Area as it pertains to various shapes. A variety of different shapes will be looked at in two dimensional and three dimensional forms and students will work in groups to determine the Surface Area of each. The concept of nets will be explored.

Preparation: Students will spend the first brief portion of the class listening and discussing. They will then move into groups of three or four where group participation and exploration will occur. For this portion, various right rectangular and right triangular prisms and right cylinders will be handed out for tactile exploration and Surface Area Determination.

Introduction: A quick review and class discussion will be given of what various geometrical forms are. From here, areas will be reviewed and the concept of surface area will be developed and discussed.

Body: Splitting up the class into groups of three or four, the students will have some time to explore geometrical shapes (specifically right rectangular and right triangular prisms and right cylinders). Surface Areas for individual shapes will be discussed and determined. Nets will be created and discussed and their various uses will be determined.

Closure: Questions and discussion will be encouraged and the surface areas of the various shapes will be understood. Students will be asked to study shapes at home and to create a net for an object that they find at home that matches one of the shapes studied. These will be compared and discussed next day.



Hands-on Pythagorean Theorem

Rationale: To give students a review of the areas of various shapes from a different perspective, leading to a hands-on look at the geometry of the Pythagorean Theorem from its origins, without calculators or notebooks. It is hoped this will be an informative and enjoyable lesson.

Instructional Objectives: Each student will be able to understand various geometrical methods and specifically those of triangles through group work with knotted strings and ‘human triangles’.

Preparation: Students will spend the first portion of the class learning about the origins and the theorem itself. Afterward, students need only to clear space for themselves and be able to freely make use of the knotted strings in their groups of three or four.

Introduction: A quick review on the chalkboard with student involvement of previously performed geometry (and the origin of the word). The basics of the Pythagorean Formula will then be illustrated.

Body: Splitting up the class into groups of three, the students will have some time to explore geometrical shapes with their strings that include numerous knots which are each spaced 10 cm apart. It is hoped they will think of geometry as a Pythagorean might have with the same tools available. Questions will be asked and encouraged throughout the lesson amongst the groups.

Closure: The lesson will be concluded with human triangles, beginning with a right 3-4-5 triangle, built with people standing shoulder to shoulder. Approximations will be made and triangles will be balanced between those who are broader shouldered than others. Questions on the activity can conclude the lesson and summarize things that weren’t clear.



Volumes of Shapes

Rationale: Students will understand the concept of volume and will be able to make comparisons to Surface Area and contrast their differences. Students will also be able to use the concept with respect to their group projects.

Instructional Objectives: Each student will understand the concept of Volume as it pertains to various shapes. The same variety of different shapes from the pervious week will be looked at in two dimensional and three dimensional forms and students will work in groups to determine the Volumes of each. These volumes will be compared to the corresponding Surface Areas through discussion.

Preparation: Students will spend the first brief portion of the class listening and discussing. They will then move into groups of three or four where group participation and exploration will occur. For this portion, various right rectangular and right triangular prisms and right cylinders will be handed out for tactile exploration and Volume Determination (much the same as for the Surface Area lesson). Students will then move into their groups where they will continue to prepare their group projects. Materials for this will need to be provided.

Introduction: A quick review and class discussion will be given in regard to various geometrical forms. From here, the concept of volume will be developed and discussed. The students will then move into groups.

Body: Splitting up the class into groups of three or four, the students will have the bulk of class time to explore geometrical shapes (specifically right rectangular and right triangular prisms and right cylinders). The volumes of individual shapes will be discussed and determined. Questions will be asked of each group as necessary to encourage further exploration.

Closure: Questions and discussion will be encouraged and the volumes of the various shapes will be fully understood. The students will then continue to work on their group building projects and specifically discuss how the volume of their rooms will be determined by their shapes and what result this will have on the heating of their buildings. This will be included in their group presentations.


b) House Building Blueprints Project The class project that will be worked on throughout this unit will be the basic design of a house or structure from the past, present or future (creativity will be encouraged throughout the project). This project will be completed in groups of three or four with the group presenting their 2 or 3 dimensional final product to the class. The project must include varying shaped rooms and structures with measurements, with at least one right angle triangle being featured. Floor areas of rooms, surface areas of walls and roofs and internal volumes for heating considerations must be determined and presented.

Individual Expectations and Grading

Three journal entries will be given (one for each class dedicated to the project work). Other entries from additional individual and group work will be encouraged. Each entry should include successes and drawbacks that were encountered throughout the project and how each related to the concepts that were learned in class. The role each student accepted and what was accomplished as part of the larger group should also be discussed.
These entries will be given a mark out of 6.

One final journal entry will be given after the student’s group presentation. This entry will include summary information on what was discovered as part of the group’s presentation. Again the successes and drawbacks of what was found during this presentation will be encouraged to be listed and described, specifically regarding unit concepts. The relative success or failure of the rest of the group presentation will not be considered here – only the observations of the individual student.
This summary entry will be given a mark out of 4

The Total Mark for all journal entries will be out of 10 and will comprise 2.5% of the final grade.

Group Expectations and Grading

Students will be expected to perform well in cohesive groups. Within each group, the roles of each student will be chosen and agreed upon with regard to the final presentation. Students will work together to explore concepts and how they relate to their specific project. Again, creativity and participation by all will be highly encouraged.
This group work will result in a mark out of 5

Students will present their blueprints and/or three dimensional projects. Number and size of rooms will be discussed. Considerations for surface area and volume of a variety of different shaped and sized rooms will be necessary for full marks. Also, as specified, there must be at least one right angle triangle featured with use of the Pythagorean Theorem in determining its size. Presenting skills will be encouraged but not evaluated.
This group presentation and final product will result in a mark out of 5

The Total Mark for each member of the group (each members mark will most likely be the same) will be out of 10 and will comprise the other 2.5% of the final grade.

Group Project - Recreational Mathematics

Martin Gardner
Martin Gardner was born on October 21, 1914. A prolific writer, Gardner has published 70 books to date on topics including: recreational mathematics (our interest here), pseudoscience, magic, philosophy, literature, and even a couple works of fiction. His most recent work of fiction is a story about Oz involving Klein bottles. Gardner is most fondly remembered for his column Mathematical Games, which ran in Scientific American for 25 years.

Gardner has no formal mathematics training outside of high school and even had a hard time passing his high school calculus course. His background is in philosophy, which he studied at The University of Chicago. When asked what he likes about mathematics Gardner responded: “There is a strong feeling of pleasure, hard to describe, in thinking through an elegant proof, and even greater pleasure in discovering a proof not previously known.”

After serving as a yeoman in World War II he became a contributor to Esquire magazine. His first submission was a short story entitled “The Horse on the Escalator.” From Esquire he became assistant editor of the children’s magazine Humpty Dumpty (Humpty was the head editor). After submitting a piece on flexagons to Scientific American in 1956 he was asked to head a monthly column focusing on recreational mathematics.

Gardner claims that the number of puzzles that he has invented could be counted on one hand. He could often be found scouring Manhattan bookstores for books of recreational mathematics that he would use for inspiration. Gardner believes that his column was so successful because of his lack of experience. While writing the column he would also be solving the problem for the first time.

In 1981 Gardner retired from Scientific American to focus on his other passions, mainly debunking pseudoscience. Gardner inspired many a future mathematician with his column which lives on in various collections that have been published since his retirement.
The problem will be presented in class.
Thoughts on the project:
This project is a good way to show students that math problems exist in popular culture and can be done just for the fun of it. It’s also a chance for students to pick a problem of their own choosing that will hopefully be a fun challenge for them and not just another problem where they aren’t interested in the answer (except that it’s correct). One of the great things about the problems in Scientific American is that they are phrased in a way where you are actually interested in finding the answer. During class it’s hard to make questions that are interesting for students, and showing them that they can actually find math problems that are “fun” will hopefully improve their opinion of math and willingness to engage in the learning of it. While I was finding the solution to the problem, I wondered how easily I would be able to find the solution if I were still in high school and didn’t have a degree in mathematics. If I assigned this project to students I would worry that the problem they pick might be so hard that they get the impression that math is just something that they will never be able to understand and use fully. On the other hand, picking problems for the students doesn’t allow them to find problems that are interesting to them and takes away a lot of the benefit of the project.

I would most likely use this project as an enrichment assignment to be completed between units, since there is no specific topic that is being used or learned here, it’s simply a general interest project. It would be fun to have everyone present their project, but I don’t know if I would have the time to spend doing in-class presentations. Instead, maybe I would have them create a poster and put the posters up in the classroom. Another thing I could do if I had one class to spare is make this the basis of an in-class math fair where everyone would create a booth and go from group to group trying to solve the other problems.

To put together a math fair I could modify this project so that the end result is a booth rather than a report and poster. If the students were not in grade 12, I could adapt the project so that I choose a number of problems that I know are solvable for their grade level and allow them to choose only from my selection of problems. Another idea would be if I had a grade 12 class and a lower grade class (grade 8-9) I could assign this project to the grade 12’s, have them make a booth with tools to allow anyone to solve the problem and bring the finished projects to my grade 8-9 class for the students to work on. I could make part of my evaluation of the grade 12’s that they present the problem in a way that is understandable and solvable for a grade 8 class.

Wednesday, November 18, 2009

Reviewing my group's work

It was great to see the solutions (and attempts) provided for the Jacobian Locks problem by my team member's - Jill and Caleb. Jill's was very neat and gave great visuals of stick people in the village along with great charts illustrating the various scenarios they went through in solving the problem. She made the mathematics behind her solution very easy to follow and came up with, what appears to be, the correct solution. Well done Jill!
Caleb's solution was great to read as well from the pictures on his whiteboard. He used set theory which also provided an easy way to follow his thinking. His many notes along the way made it enjoyable to read. The "oh poo!" comment when he realized it was not working, reminded me of something I have exclaimed at that point myself.
Both of these solutions were great to read and it made me wish that my own weekend had not been turned upsidedown and that I had personally had more time to work on this problem.

Sunday, November 15, 2009

Jacobian Locks Problem Solving

The problem from page 176 in Thinking Mathematically goes like this:
A certain village in Jacobean times had all the valuables locked in a chest in the church. The chest had a number of locks on it, each with its own individual and distinct key. The aim of the village was to ensure that any three people in the village would amongst them have enough keys to open the chest, but no two people would be able to . How many locks are required, and how many keys?

I initially figured that there had to be some kind of trick to this. I knew that if every villager had a key to each lock, then there would be no way to prevent two villagers from opening any number of locks. So I began to think of tricks. First, I thought that there is only one lock and each villager has one third of a key that together form a full key. But this would mean not every lock has its own individual key. Then, I thought maybe the chest is up in the air and it requires a human ladder to reach it, but two people could not. This would require only one lock and each villager to have one key. This might be the solution, but I doubt it. Then I thought maybe the chest is very big (it does hold all the valuables for the village) and has three locks spread far around it. Each villager has one key that opens a lock which closes again right away. Three villagers could do this but not two. The same would work for three locks with three keys that break upon their use. Then every villager would need only one key, but the keys would not be distinct. Then I got to thinking that there must be something in the order or way the locks are open.

Tuesday, November 10, 2009

Two Memorable Moments from Practicum (Quick Write)

1) Teaching Completing the Square - It was my second lesson. The first went well, but was a learning experience. But, with Completing the Square, I got the AHA! moment from the class. I invoved humour and previous review and it felt good to look out at every face, most smiling, some nodding, and many getting the beginnings of a difficult topic.

2) Halloween Costume - Dress up day was the last day of the two week stretch. I had developed a rapport with many of the students and was starting to feel I was well liked. One student that had me wondering on day one, "am I going to have problems with this one" sat in the third seat from the enterance. I walked in in my calculator costume and he said, "you made my day by dressing up as a calculator". It brought an instant smile to my face. As I walked away, I heard him say to a friend, " I want to dress up as a calculator".

Sunday, November 1, 2009

Quick Writes

Divide: A word with many meanings - some destructive and othersconstructive. A noun or a verb. A divide could be an area across a valley where one might want to build a bridge. A divide could be a stretch between cultures - a social divide. Of course, to divide is the cutting of another object or number into smaller pieces. Divide is the root word of division, which is the noun that is used to claim where a divide ha been made. Divide begins with the letter D and contains two i's. No other basic mathematics skill begins with the letter D

Zero: Zero is a unique word - a word to describe nothing - truly nothing. Zero is a concept that seems obvious today, but in the early days of mathematical study, zero did not exist. it was completely left off the number line of integers. It is an amazing concept when you think of it - something that's sole purpose is to describe nothing. Zero makes adding, subtracting and multiplying very easy. But something very interesting happens with division. Zero can divide as many times as one wishes. I can give you zero things many, many times over. However, when I try to divide by zero, it becomes mathematically impossible. Why, you ask? Well to dived by zero means that I am not dividing at all, which makes it impossible.

Division by Zero

Some feel a division at the root is all bad
To seprate into groups
Breaking apart makes things lesser
And limits their worth
When seen through clear lenses
It is in all that we do
Thus division by zero should be liked most of all
For it, and it only, does not yield to divergence

Friday, October 16, 2009

Review of Teaching Geometric Series

Our team micro teaching project went well. Mina and Sam were excellent teaching partners and it was a well planned lesson (we are thankful to Sam for all his hard work). We combined three basic portions: Mina gave an introduction, Sam furthered this with a proof and further explanation and I used bouncing tennis balls to show a real world example. I believe most of our 'students' enjoyed the presentation and were able to follow the reasoning (see feedback to follow). Our weakest element was our use of time. We went longer than we had planned to and so felt rushed. It was great experience, though, and has given us good exerience teaching this subject in front of math students.

Averages on Feedback
Clarity
Structure was clear - 4.14
Verbal & Visual were clear - 4.07
Math ideas were clear - 4.14
Active Learning
I was engaged in learning - 3.57
A variety of activities offered - 3.64
Instructors showed engagement was valued - 3.93
Offered activities connected to other math areas - 3.93
Connected to other areas of life and culture - 3.43

Some comments included: good classroom management, very engaging, clear instructions, good humour, clear review, concise, fun activity,good easy examples, it's good to have a real life problem at the end, very well organized.
Also, on the not so positive: bit much time in review, short on time, the board was blocked (couldn't see), wait for students to respond, little long and drawn out, the pace is too fast, needed examples of geometric series at first, define terms clearly, relax when speaking.

Wednesday, October 14, 2009

Geometric Series

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.

thanks to Sam Douglas for this lesson plan

Citizenship and Democracy Finding a Place in the Mathematics Classroom

Probably one of the last things the average person thinks about when considering what should be taught in the mathematics classroom is social issues. There are so many branches of the tree that make up teachable mathematics, how is it possible to include citizenship education amongst them? Elaine Simmt makes the argument that it is possible and should be done.
Simmt argues that because numbers and statistics are so vital and interlocked into our daily lives, it would be irresponsible of our leaders at all levels not to incorporate the teaching of them to all who enter school. She suggests, “mathematics education and citizenship education need not be distinct tasks of the teacher; rather, appropriate mathematics teaching also prepares the student for citizenship”.
While a noble cause and one that most involved with mathematics would think an excellent new use for the age-old study, this is quite difficult in practise. Simmt does not suggest, as the reader might originally think, that debate and direct tackling of social issues is what should be taught. She argues instead that slight changes should be made in the way that teachers teach the same things they have always taught. This in turn will better prepare the future citizens of the world. It is a great goal that should be welcomed by all math teachers, regardless of how improbable it seems.

Friday, October 9, 2009

What if Not?

The Strengths of this method are seemingly quite obvious, despite being seldom used in ‘real world’ problem solving. This strategy enables the problem poser to consider a great many alternative situations to the current problem, in order to see the current problem in the clearest possible light. As with any tangible object, if one can pick it up and rotate and flip it and hold it in different shades of light, one is going to know much more about what it looks like than if it were simply stared at from across the room.

There are unfortunately limitations to this strategy. Time constraints can become a factor if a multitude of other situations are considered instead of focussing on the job at hand. Confusion and loss of focus can result for many people if they delve too deeply into other explorations. Some ‘what if not’ questions, as can be seen with some of the examples in the book, are simply a waste of time and do not help with the solving of a problem at all. Another limitation to consider is that many problem solvers may be thrown off their ‘rhythm’ by allowing their minds to wander around a problem.

In our microteaching assignment for Wednesday, we have been asked to incorporate the ‘what if not’ strategy. I think this will be a fun a way to ‘teach’ the concept of Arithmetic Series, especially considering the students will already know the concept. Questions come to mind such as “what does it mean for a series to be arithmetic and what would it look like if it were not arithmetic?”, “what if we could not generalize an arithmetic series?”, “what would a series look like that had no sum?” It should be a fun and thought provoking way to introduce a concept.

Monday, October 5, 2009

Take 10 on "The Art of Problem Posing"

This book begins with a witty and occasionally funny style of writing. I enjoyed the repeated fifth reason for why a repetition of key ideas is used throughout the book.

I question why the introduction needed to be so long and incorporate revisals from previous editions. Perhaps this could have been done as a seperate part of the book.

I like how the book can be enjoyed by all levels of mathematical ability.

I enjoyed the vast array of different questions that were given as possible topics of thought. Some I just may use in future problem solving.

I thought it was interesting to involve history as a possible exploration when thinking about problem posing.

I enjoyed the First Phase of Problem Posing being Accepting. I think this is the same first phase in addictions counceling.

I enjoyed each of the five examples for Sticking to the Given.

I found it interesting that two offbeat examples of questions were given at the end of the Supreme Court Judge handshake example.

I am looking forward to the second phase of problem posing.

I am still left wondering if being better at problem posing will help my abilities (and those of my students) in problem solving.

Friday, October 2, 2009

Two Future Letters

June 30, 2019

Dear Mr. Whitbread,
Thank you for the time you took to teach my Math class over the last school year. I began taking your class with a dislike for Math - I had had previous bad experiences. I'll never forget your first class where you encouraged us all to drop our fears of math and learn again to appreciate the beauty of the subject. You told us that most of us would never love everything about math, but all of us could learn to love certain aspects - much the same as all our other subjects. I liked that and it has stayed with me. I will always remember how I looked forward to your classes. Sure there were boring parts of classes, but you always gave me something that would make me think.

Dear Mr. Whitbread,
After spending a large number of hours in your classroom, I can't help but look back and think it was a large waste of time. It wasn't all your fault - math is just boring and I could never understand why we needed to learn it. I did everything I could to not fall asleep in class while you were working on examples. I really wish you hadn't called on us for examples and made us do group work. It made me feel stupider than I already had felt I was when I didn't know the answers to your questions. Why couldn't you just teach us like my previous teachers and make us learn things at home? It would have made me feel way more comfortable.
Maybe you can be a better teacher in the next ten years.

While I hope more letters conform to the first, I do expect some negative feedback as well. I know I am going to be teaching a subject that carries a lot of dislike by a great many students (and people in general), but I hope that my positive, good natured demeanor is able to win most over. I hope I can help those that already love mathematics to continue to learn its graces.

Dave Hewitt Video

Our MAED 314A class got to watch a short video featuring two lessons being taught by Dave Hewitt, a grade 8 and 9 Math teacher in the UK. The enjoyable part of watching Mr. Hewitt teach is that he seems to make the learning very enjoyable for his students. There is a definite theatrical style incorporated into his teaching method that seems to hold the interest of everyone in the classroom, and those watching the video. The idea of using a meter stick to make a tapping sound and to use space on the walls beyond the chalkboard are two tools that he uses to keep the students engaged and actually learning in a non-boring, ‘outside the box’ way. Once he has established the students basic understanding of how numbers fit on the number line, he begins to introduce algebra without anyone realizing variables have been introduced. When many math students see their first x or y as part of a mathematics problem a wave a fear goes over. The students in Mr. Hewitt’s class had nothing to fear when he was simply “thinking of a number that…” When an x was then introduced, it was a seamless progression. It is entertaining and engaging methods such as those of Mr. Hewitt that I think would make the Secondary Mathematics classroom a much more enjoyable place for so many young minds and I hope to teach in that classroom.

Wednesday, September 30, 2009

“Battleground Schools”, Summary/Reflection

“Battleground Schools”, by Susan Gerofsky is an interesting read that discusses what styles of mathematics are taught, and historically have been taught. Political lines have been drawn since long before we had formal education, continuing to this day, and it is unsettling to know that they clearly do not stop at the entrance to the classroom.
Most people are probably quite oblivious to the idea that there are different ways to teach mathematics. Surely, it seems quite standard to the layperson; there are numbers and letters and equations and identities and if you put them together in a classroom you get math. While this is true, most people can probably also recall different math teachers that taught them as students and the different methods that they used. Some of these teachers relied heavily on memorization work to learn concepts and, hopefully, some used different types of methods so that concepts were understood. I am sure it would be found that those exposed to the latter would have a better appreciation of the beauty of mathematics.
As stated in the essay, “mathematics education has oscillated between two poles“; that of “progressive and conservative”. These political stances are quite clear outside the North American classroom – you are either liberal or conservative, democrat or republican with little agreement between the two sides. It is upsetting to find these stances have serious impacts on a child’s learning. Gerofsky points out “few progressivists would argue against some necessary degree of fluency in basic mathematical procedures, and few conservatives would be as radical as to advocate fluency exclusively to the point where understanding would be discouraged.”
“Battleground Schools” discusses a political/educational issue where, like formal politics, few are comfortable discussing. There seems no end to the conflicting decisions that will be made regarding mathematics in the classroom. One can only hope that individual teachers are finding a balance and teaching mathematics so that not only will their students learn concepts, but they just might start to enjoy them.

Monday, September 28, 2009

Reflection on Teacher and Student Interviews

My group, also consisting of Amelia Landon and Sam Douglas, took on the task of asking a teacher and a student five burning questions in an effort to discover teaching styles and methods that will work well for us. We were able to set up an interview with John Yamamoto, the Secondary Practicum Program Coordinator for the Education Faculty at the University of British Columbia. While he is not currently a secondary teacher, he taught for many years in Trail and Burnaby, in the regular, correctional and special needs school systems. Our current high school student was Sam’s younger sister who attends grade ten.
The interview with Mr. Yamamoto was very interesting as he has seen many different teaching environments over his career.
Our first question for him was, “what did you find to be your biggest challenges with your early teaching experiences?” He discussed his early issues with classroom management and working around the noise of the classroom as opposed to controlling it. He discussed his solution was to make the expectations of his classroom as clear as possible.
Another good question was, “what accommodations have you made to help students with learning difficulties?” This question struck a chord considering his diverse previous experience. While working with the deaf, he was forced to consider the pace at which he spoke, especially when working with a translator. Although, he said with practice this was not an issue at all. Working in the correctional system, he was forced to deal with many personality issues and said in all school systems, “their problems usually have nothing do with you (as the teacher)”.
Our final question asked Mr. Yamamoto what he enjoyed most and least about teaching. He said he loved the interactions with the kids and enjoyed watching them grow, but it was always tough to see them move on to new phases in their lives beyond high school. He also indicated poor balance in the curricula and teaching large class sizes as things that made teaching difficult, but he said overwhelmingly that the good outweighed the bad.
The interview with our grade ten student also provided some interesting responses. She was asked how her first impressions of a teacher are formed. Her answer listed a good sense of humour and a healthy level of classroom organization as being important for making a decision.
Our grade ten student was also asked how she felt about incorporating more group activities in Math class. She was not fond of the idea saying that she prefers to work alone and that group work can lead to confusion. While some students do like group work, it is good to note that there will be many who will feel uncomfortable.
Finally she was asked about a memorable way that a teacher has helped you to understand a tricky concept. She relayed a funny story used to help the understanding of solving an equation. It is nice to hear that sometimes a little ingenuity and humour can go a long way with adolescents.
There will never be a universal agreement on which is the best teaching style from either the teacher or student perspective, but it is very helpful to note things that tend to work and be well liked. While only interviewing one teacher and one student, this was a great exercise to find what some perspectives are on both ends.

Reflection on Group Interviews

An eventful day of group presentations has given us some reports on the real world thoughts from both the teacher and student perspective. These reports included answers from teachers who loved their jobs and also those who did not, it seemed. They included stories of teachers who loved teaching in a 'new-age' classroom with alternative media usage and students teaching themselves to some degree. There were also teachers who liked teaching in a very traditional classroom using traditional methods.
From the student perspective, there were reports of those who loved mathematics, and their teachers who taught that subject, and those who did not. There were those who enjoyed learning in a traditional classroom setting and those who liked learning with group and modern teaching methods. This began to sound familiar.
It seemed that the only thing that stays constant across the outlook of both the teacher and student is that there is no such thing as a universal perspective. People will think different thoughts about different areas and the methods of no one teacher will be liked by all. It makes me think to my own future teaching days where I will simply bring to the table those methods and strategies that I feel are most effective and try to keep most of the people happy, most of the time.

Wednesday, September 23, 2009

Response to “Using Research to Analyze, Inform, and Assess Changes in Instruction” by Heather J. Robinson

It seems clear that the image of the teacher standing at the chalkboard or overhead projector lecturing for an hour, or at least its effectiveness, is dead. However, in thinking back to some of my own teachers that I had in high school, this idea should have been dead a long time ago. I believe many teachers have known for a very long time that their teaching methods were largely ineffective; that they were only ever reaching a small percentage of the students. The others would have to pick it up on their own or through tutors. If these teachers had done some research into their own teaching methods as Robinson suggests, it would have been easy to see where they were failing.
I liken teaching without doing self research to a workout plan where nutrition is neglected and results are never studied. If a workout schedule of any kind were followed like this, success would surely be minimal. To not constantly be reviewing your methods and testing their effectiveness is to not complete the job of being a teacher. In order for learning to occur across the classroom, with strong students as well as weak, effective teaching methods must be adopted. It is the individual teacher that must figure out what these are.

Two Memorable Math Teachers

Mr. Fraser – I will never forget sitting in a grade 12 math class with the legally blind Mr. Fraser at the front. I was sure at the beginning of the year that students would be taking advantage of his easy going personality and his blindness to cheat on exams and assignments - very few ever did. I believe this was because of the incredible amount of respect that students had for the man. His large amounts of time that he devoted to helping students understand concepts after class and the respect that he showed for each and every one of them is something that I will never forget.

Mr. Ritchie – Grade 9 math was very enjoyable thanks to Mr. Ritchie. His soft spoken and easy going nature made learning fun and it was an approach that I had not seen previously. His keen interest in extra curricular activities and engaging conversations in the classroom made him a very memorable teacher.

I will try to implement things from both of these great teachers. They both had similar, easy going styles that were well liked by students and I hope I can use mine to be as good of a teacher as they.

Monday, September 21, 2009

Self Assessment on Microteaching: How to Juggle in 10 Minutes or Less

My first teaching assignment for MAED 314A was a fun one that was well received by my group of five.
My BOOPPPS style lesson plan (see previous blog) was followed closely and I managed to fit the lesson nicely into the ten minute time frame. I was happy to find that no one had previous juggling experience. This would have made for a slight complication, but one that I would have had a plan for. I was also happy to see that each of my four students (Amelia, Mina, Nathan and Sam) was, or at least pretended to be, keenly interested in the activity. In the beginning – using one ball and then two - it was rewarding to see each of the students making progress with the skills. I did my best to give good feedback as this occurred and made suggestions where needed.
I received great feedback from my “class” which included, “great demo”, “great step by step instruction”, good “starting easy to progressive to hard” and “good use of descriptive terms”. These were very nice to hear and made it feel like my teaching session was a success.
The constructive feedback I received was unanimously to use something other than tennis balls for this lesson as they were too bouncy. I agree completely with this recommendation and would try to use bean bags or something similar to avoid the balls bouncing away from those learning the skill. I was also able to pick up some ideas for how I would instruct in the future that I hadn’t thought of when I planned the lesson. Amelia noticed midway into the lesson that when jugging three balls, the beginning hand had to contain the two balls and not just one. This would be highlighted in a future session.
Overall, this microteaching assignment was a success and was seen as rewarding and confidence building to myself and to those I was trying to teach. I can only hope that I teach classes in the future that are as interested and well behaved.

Friday, September 18, 2009

How To Juggle, in 10 minutes or less

Microteaching Assignment Sept. 18th, 2009

1) Bridge – Introduce simple 2 and 3 ball juggling with tennis balls (approx 1 min.).

2) Teaching Objectives – To introduce students to the art of juggling through display and participation exercises.

3) Learning Objectives – Students will be able to understand the movement of the balls in basic 3 ball juggling and will be able to master the skill through their own practice after the session. Students will understand that juggling cannot really be learned in 10 minutes, but it was sure a catchy name for the task.

4) Pre-test – Students will be asked if any have previous juggling experience (approx 1 min.). In this case, exercises will be set up to practice two person juggling, under the leg juggling or 4 ball juggling.

5) Participatory Activity – Students will each receive one ball and practice the simple tossing from one hand to the other and back again, working on keeping the ball at an ideal, consistent height (approx 1 min.).
Then, another ball will be introduced and students will practice “one, two, nothing to throw”. (approx 4 mins)
Those that feel comfortable can then try with 3 balls and those who do not can continue with two ball juggling with one or two hands. (appox 2 mins).

6) Post Test – Ask students how they enjoyed the activity and ask each to display the new skills they have learned. (approx 1 min).

7) Summary - Reiterate to the students that much additional practice will help them attain the skill. Thank them for their time and efforts and wish them well in their circus careers if teaching does not work out.

Thursday, September 17, 2009

Commentary on Richard Skemp's "Relational Understanding and Instrumental Understanding"

As a former mathematics student who plans to teach mathematics in the days ahead, I found Richard Skemp’s essay, “Relational Understanding and Instrumental Understanding” very interesting. Skemp introduces the idea of a difference between these two types of understanding, by relating the idea of “Faux Amis” or false friends – words in two different languages that look very similar but have different meanings. Skemp claims, “I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’”. It is interesting to realize this statement is something I had thought about in the past, but only as it related to different learning styles. I am sure anyone who has studied math at any serious level has identified themselves as someone who understands relationally, although most, if not all, have used many instrumental ‘tricks’ along the way.
Skemp goes on in the article to take the “Devil’s Advocate” position in an attempt to understand the benefits of Instrumental Understanding (It is clear early on that Skemp favours Relational). Under this position he states, “instrumental mathematics is usually easier to understand”, “the rewards are more immediate”, and “one can often get the right answer more quickly and reliably by instrumental thinking”. He counters his own opposing stance by claiming relational mathematics “is more adaptable to new tasks”, is easier to remember”, “it can be a goal in itself”, and its “schemas are organic in quality”.
Skemp continues with good reasons for why the two types of understanding should be understood by math teachers and the benefits of both in different situations. Skemp’s article highlights a very interesting theory behind why the teaching of mathematics may cause its dislike for so many people, and not the mathematics itself.