“Math Projects” Assignment

(#6) Math 10 Project: Geometry of Design

group: Michelle, Meghan, Paul

This is certainly an interesting project from the point of view of math appreciation.  There are many students who are turned off by algebra and numbers, and for them this playful visual exercise can do a great deal to draw them in to the world of math.  The practicality of this project may be called in to question, as there is notoriously short time in class.  However, this may be a criticism of the school’s scheduling more than the assignment itself, which is wonderfully enriching and deserves more time than it will inevitably be allotted.



Strengths:

• Requires very little base knowledge.  This makes it accessible to everyone in the class.  It may inspire the struggling students to see a project in which their previous weaknesses in mathematics are not compounded.
• Combines math, history, nature and art.  Let’s face it, most people find art and history far more interesting than mathematics.  Showing how math can relate to the these “fun” subjects will go a long way towards showing students how enjoyable math can be.
• This project enables students to choose their own piece of art or object from nature, which allows for broad ranges of interest.  
• After students complete this project they will see how mathematics is in so many different aspects of everyday life that they probably did not realize before.  One of the main outcomes of this project is math appreciation.   

Weaknesses or Constraints:

• Students may be able to find geometric shapes in well known art on the Internet.  This will prevent the student from playing around with shapes and tracing paper to find geometry in the item themselves.
• Perhaps the topic might be too narrow and not really interest all students.
• Project will be hard to cram into a busy curriculum as it is time-intensive and does not cover many PLOs.

Possible Modifications, Adaptations or Extensions:

• One possible modification of the project is having the students orally present the steps they took to finding geometry as opposed to the step by step write up. This might benefit some students, that  are more comfortable verbalizing their process than writing out the steps. Another advantage is students will have a chance to share their findings with their peers.
• Perhaps students could then make their own piece of art work which is based on geometric shapes or mathematical ratios.  

Our Project:

Math12 Project: Mathematics in Art Design

Grade Level: We chose to do this project in Math 12 because geometry is no longer in the curriculum and this is a great enrichment project for these math students who have knowledge about geometric and trigonometric and shapes.
Our project will span two classes over a period of one week.

Purpose:  The purpose of this project is for students to understand and see how mathematics is within many works of art, even though it is not obvious.  The planning of many pieces of art includes mathematical shapes, calculations and ratios.  Students will play around with and sketch geometric shapes, the golden rectangle, logarithmic spirals etc. and see how they can be manipulated to form interesting compositions.  Students will develop their own work of art which is based on their chosen mathematical elements. 
    

Description of Activities:

Day 1:   

• Show students examples of famous art which contains and is based on geometric shapes or ratios.  There will be a PowerPoint presentation or overheads with examples and also a hand-out for students which includes: the Mona Lisa, the Parthenon, Egyptian art, Mondrian, daVinci etc.  
• After students see how these basic shapes and ratios are behind the planning of many works of art students will learn how to draw, using a compass and ruler, these basic geometric shapes.  Some shapes they will practice drawing are the golden rectangle, logarithmic spirals, polygons, stars and possibly trigonometric functions.
• Students will decide by the end of this class which geometric shape(s) or ratio(s) they want to use to construct their artwork drawing.  

Homework:  Students plan and sketch their artwork drawing.  


Day 2 (One Week Later):  

• Students make their drawing on large poster board.  The drawing can be done in markers, charcoal or pastels.  By the end of the class students should be able to have their work finished.  If the project is being done in the art room then students could use other materials such as paint, collage, etc.  

Sources:
Kimberly Elam (2001). Geometry of Design: Studies in Proportion and Composition. (NY: Princeton Architectural Press.)
Miranda Lundy (1998).  Sacred Geometry. (NY: Walker & Co.)


Timeline of Project:  Two classes separated by a week.  Students are required to plan their artwork between the two classes and come to the second class with a complete plan so that they have the maximum amount of time to complete the artwork in the second class.



Students are Required to Produce:  

• a large composition drawing which incorporates at least one mathematical element in the planning.  
• all their planning materials such as sketches of shapes, rough drafts and rough work of their final composition.
• A written or spoken explanation of the mathematics involved.

Marking Criteria: Rubric (20 marks total)

Use of Class Time (3 marks)

• 0 - Absent
• 1 - Come to second class without plan, make poor use of class time
• 2 - Come to second class without plan, make good use of class time
• 3 - Come to second class with plan, make good use of class time

Presentation (2 marks)

• 0 - Unfinished or missing
• 1 - Poor effort, obviously done at the last minute
• 2 - Decent to good effort.  Most students will receive this mark.

Mathematical Content (10 marks)

• 0 - Missing
• 2 - No apparent mathematical content
• 4 - Mathematical concept attempted but incorrectly applied (such as an unmeasured freehand spiral)
• 6 - Mathematical concept properly applied, but with errors in explanation.
• 8 - One mathematical concept properly applied and explained
• 10 - More than one mathematical concept properly applied and explained

Completeness (5 marks)

• 1 mark for each of the following completed on time: Planning materials, rough draft, explanatory piece
• 2 marks for final draft submitted on time

Justification for Rubric: Since this is a math class and not an art class, the artistic component is negligible.  The largest component is the mathematical content.  It follows a linear progression where 0 marks are earned for doing nothing, and 8 marks are earned for doing the minimum.  An extra 2 can be gained by going a little bit beyond.  The students would, of course, be made aware of this beforehand.  It is expected that most students will attempt to incorporate 2 mathematical concepts, and even if they completely mess one of them up, they’ll still achieve 8 out of 10.  We feel that marks should not be difficult to obtain on enrichment projects such as this.

Response to Creativity, Flexibility, Adaptivity and Strategy use in Mathematics

The story at the beginning of the article really emphasizes the importance of adaptability.  It was great that this student who normally struggles with math was proud of his innovative method to solve the problem, however, I feel he has missed a very important concept of mathematics.  Ferit failed to see, or was not lead to see, the many methods and strategies that must be employed to similar problems.  He believed that since this strategy worked for a particular problem, that it must work for all the other problems like it.  
A key element to problem solving is recognizing which strategies work best and employing those first. The four words in the title of this article, creativity, flexibility, adaptivity and strategy, are very important in terms of problem solving.  Students must not be afraid to get creative.  Even though they may very well chose a method that leads to no solution, they always learn something in the process.  These students who use creative problem solving methods will not only find unconventional ways to solve problems but develop their creative aspect which will benefit them in all areas of their future.  Students must be flexible when attempting mathematics.  The teacher can be a great source of inspiration for being flexible. 
I feel that it is quite true about students who have more previous knowledge are less able to be flexible and adaptable within mathematics.  I feel that rather than constantly building on concepts, problem solving requires students to forget what they know and start from scratch.  I feel that students who will be more successful are the ones who try something different and are not stuck within the constraints of previous concepts and methods.  But how do we as teachers get students to do this? How do we help students tap into this creative side and loose their fear of getting stuck.  I feel that we must lead by example.  When evaluating problem solving we must place value on the thought process behind it and also place value on all the methods they tried which did not lead to a solution.  I think that the 'Two Column Problem Solving' type assignments are a great way to emphasize the importance of creativity in mathematics.    

Two Column Problem Solving

Word Problem Analysis

You have a four-wheel drive all-terrain truck, and you need to make a 27 000-km trip. Each tire can be used for 12 000-km. The four tires of the truck are new, and you have five new tires in the back of the truck. How can you use the nine tires to complete the trip?

(MathPower11 Ch5-Rational Functions, Equations and Inequalities)

Practical?
This problem is not very practical. Although, it deals with tires on a truck and making a long trip, it has this really weird aspect of having to change the tires during the trip. The reason this problem is odd is because no one usually drives long enough in a single trip for new tires to wear out. Say you drive for eight hours a day at 100km/hour it would be over 33 days of driving. I feel that this problem would have a more practical aspect if it was a semi-truck because they may drive this many kilometers. But changing the problem to a semi-truck brings the problem to eighteen tires and not four being used at one time, so maybe this makes the problem more complicated than intended.

Memorable?
The only thing memorable about this problem is wasting time on a trip to constantly rotate and change tires on the shoulder of the highway.

Solvable?
Yes this problem is solvable in more than one way. There could be the issue of which solution is most efficient, meaning the least number of stops to change tires.

Can there different positions or interpretations?
As being someone who has car knowledge, I have an issue about driving for extended periods of time and at high speeds when the tires have different levels of wear. This is not only bad for the car but also not so safe. If the car was only front wheel drive or only rear wheel drive then we could have the two front tires at the same level of wear and the two back tires at the same level of wear. But with this problem, the truck is all-wheel drive so the tires should all be the same so the tires are balanced.

How would kids interpret it?
I think that kids would interpret it correctly. I don't think that there would be any major misunderstanding about what the problem is asking.

Is this problem strange?
Yes! This problem is quite strange, although there I'm sure exists stranger problems. This problem does relate to driving and lifespan of a tire in kilometers. I think this problem should not be worded as "you" have a truck... but rather that someone wants to make this trip and they have these tires etc.

Rewrite or Extend?
Since I have knowledge about cars and changing tires I would add to the question that the tires need to be as balanced as possible. Also, students may not be familiar with any of this if they do not drive or are not in automotive class.

Practicum Stories

On my last day of my practicum I taught a Math9 class where we did a review of exponents and unit quiz. I spent approximately half the class going over questions they had from a quiz which they got back. Then the students were given 20 minutes to review and independently study for the quiz. I was not sure how this would go because they are a younger grade and I wasn't sure if they would stay on task studying for the quiz. However, I wondered around the room and answered questions, and the students took advantage of this time for the most part. My sponsor teacher had said that the students needed to do the quiz that day because they were already behind from the other Math9 classes, but they also may not have been completely ready for it. He had suggested that we do a group quiz or something along those lines where they could collaborate with other students.
I came up with a type of group quiz, but one that everyone handed in their own papers and not one from the group. Students were given 20 minutes to work on the quiz by themselves, but I told them that they would have a chance to get help from other students after. They were quite excited about this, but I held off telling them the details. After 20 minutes they were each given 3 slips of paper with their own names on each. These were their "lifelines" where they could get help from 3 different students. When students asked another for help, they would give the helper their slip of paper. So as each students handed in their quiz they would also hand in the slips of paper of the students they helped. However, each student could only help up to 5 people, which prevented the top students getting bombarded with everyones questions.
This group quiz went really well. My sponsor teacher was nervous and excited about doing something like this with the class because he had never done anything like this before. He is a more traditional lecture style teacher and he wants to start employing more creative activities in his classes in the future. He said that he would most certainly do this type of group quiz with his future classes.

I had another great experience with a Math11 class in which I introduced polynomial functions and properties of their graphs. I took the class to the computer lab and had them use the graphing program graphmatica to explore different graphs of polynomial functions. I chose to use the computers rather than graphing calculators because I felt they were more user friendly. I personally had a hard time figuring out the calculator in high school and I didn't want the lesson to be focused around how to use the calculator. I made a worksheet with linear, quadratic and higher order polynomial functions and a chart had to be filled in by the students. The different headings were: shape of graph, degree, leading coefficient, domain, range, minimums, maximums, x and y intercepts.
I was thrilled with how well the lesson went and how involved the students were graphing the functions. My sponsor teacher was very happy with the outcome and will be using the computer lab for graphing in the future. He had never used anything other than the graphing calculators for lessons like this.

Micro-Teaching Feedback Reflection

We were fortunate because we were able to do our lesson with the entire class. One advantage is that we were able to get more feedback. Another advantage is that I experienced the difference in classroom management from a smaller group compared to the entire class. I had a bit of difficulty getting everyones attention at the end. This showed me that I must be more assertive and speak louder, and simply waiting does not work.
We decided to do a jigsaw activity and one major advantage I realized is that it takes the pressure off the teacher. Rather than the teacher having to plan a lesson covering all the material, the students take over the teacher role and teach each other. The teacher still should walk around and engage with students but it was nice to have a break. Also, the students are more actively listening and learning at each group. They knew they had to go back to their triad and teach the others so this caused them to pay attention.
We should have allowed more time at the end for them to teach each other. We received a few comments on the feedback sheet stating there was not enough time here.
One area on the feedback sheet we received a low mark was connecting our topic to other areas of math. We perhaps could have brought this topic more broad and tied in other ideas.
We had many positive comments about using history (Babylonian, Greek), visuals (ancient tablet, cuneiform script) and relating to real like (base 60 numbering system). I really enjoy history and I worry with a lesson like this because some people find it dry, but I was pleased to see that some comments said that we made a dry topic interesting.
An area I could improve with my group was to involve everyone more since my structure was more lecture type than the other two. I asked for volunteer and when people hesitated I hesitated too! I should have had thought of a different method of engaging them if no one volunteered.
Overall, I am really pleased with out micro-teaching lesson. I have only seen this jigsaw activity in my other classes with humanities related topics or reading. I see now that this can work in a math class. But I must remember that there will not be 3 teachers.

Group Micro Teaching – Apprenticeship and Workplace 10

Bridge: 2 min

Catch interest by having students form groups of three and numbering themselves 1,2,3 for jigsaw activity in which they will become the expert on some aspect of the history or modern use of the Pythagorean Theorem.

Learning Objective:

Students will be able to demonstrate an understanding of the Pythagorean Theorem by describing historical and contemporary applications of it.
> To learn a historical use for Pythagorean Theorem and to understand where Pythagoras' inspiration came from.
> To understand that this theorem came about from a practical application, not pen and paper theory.

Teaching Objective:

> Learn and improve our methods in conducting an engaging and educational lesson that will reach everyone.
> To interactively show the motivation for the Pythagoras Theorem, introduced to Pythagoras by the ancient Egyptians.
> To try out the Jigsaw strategy.

Pre-test: 1 min

> Who can tell me what we already know about the Pythagorean Theorem?
o Looking for formula, triples, and to see if anyone might already know where it comes from.

Participatory Learning: 7 min

> Split groups into 1’s, 2’s and 3’s.
o 1’s at Egyptian Station (Deb)
o 2’s at Babylonian Station (Michelle)
o 3’s at Modern Application Station (Nadine)

Station 1 – Egyptians
> Explain how ancient Egyptians used a right-triangle to redistribute fields after yearly flooding.
> Explain how they made a right triangle using rope and knots
> Explain how Pythagoras became involved/interested
> Make a right-triangle with a piece of string and ruler. Make a 3, 4, 5 triangle and see if they can find a right angle in the classroom with it.

Station 2 – Babylonians
> Explain Babylonian Mathematics and history of Pythagorean Theorem.
> Teaching Babylonian numbering system and ancient tablet.
> Briefly cover other ancient cultures in which Pythagorean Theorem was known and used

Station 3 – Modern Uses
> Talk about how surveyors use their equipment to set up right hand triangles in order to calculate distances etc…
> Have students using laptops research other possible modern uses for Pythagoras.
> If they cannot find any, have them brainstorm a list of possible fields that they think may use the Pythagorean Theorem.

Materials Needed:
o String and rulers
o White board, diagrams of tablet and Babylonian numbering system
o Laptops, paper to write ideas down on, pens

Post-Test: 3 min

Groups of three re-form and students teach other members about their area of expertise. Only have 1 minute each to share.

Summary: 2 min

Students complete the statement: The one thing I learned today that I didn’t know before was _______________________________________________________________.
(Orally if time allows.)


Group: Deb, Nadine, Michelle

Mason, Thinking Mathematically. Ch 2&3 Response

The three phases of work are explained very thoroughly by Mason in Chapter 2 of his book Thinking Mathematically. I was struck by how detailed and structured each phase is. These three phases would be very beneficial for students in our classrooms to understand. I think that the entry phase would be particularly useful for students to further develop their skills. If the student does not plan out their attack first, then they will most likely to get stuck. However, if a student asks the three entry phase questions then they are most prepared to enter the attack phase. The first question (what do I know?) mostly involves organizing and categorizing data. This will greatly help students to know what data they have to use and also make sure that they use all the data. The second question (what do I want?) seems quite obvious but I feel is very often missed. Without fully understanding this vital point of the question, then the student spends unnecessary time working in the wrong direction. The last question (what can I introduce?) is great for students to realize this is where they can use their creativity and try different methods. Some areas to consider under this question are notation, organization and representation. After reading these two chapters I still believe that the entry phase would be the most important to spend time on with students. Too often this phase is rushed or skipped. I feel that this phase is the one that gives students most problems and causes them to get stuck.

Forbidden Territory

Mathematics is anything but unknown.
There are reasons, proofs, logic and solutions for it all.
Not only can answers be found, answers can be proven.

However, there is one little situation that renders unknown.
There is a black hole surrounding this dilemma.
It’s hard to believe because there are only two simple components.

The first component is divide and the second is zero.
Apart these two terms pose no issues or threat.
But if arranged one on top the other, this is forbidden territory.

There lacks a simple way to explain this phenomenon.
Inconsistency, irreversibility, indefinability and asymptotes.
Dividing by zero, one of the few things which math cannot do.

Timed Writing

DIVIDE

• Splitting and putting into smaller groups
• cutting something up unto smaller parts
• It could be a land into countries or a pie into slices
• a type of organization or grouping
• mathematically divinding- long division, fractions
• boundaries or lines
• smaller parts which all have something in common.
• smaller parts are different sizes or the same size
• sometimes division is not into nice whole numbers.
• dividing according to weight, value, size
• opposite of multiply
• faster than subtraction

ZERO

• nothing, none, zilch, absence, no items
• greater than -1 but smaller than 1
• in the middle of the negative numbers and positive numbers- divides the signs
• cannot divide by zero
• anything times zero is zero
• 0
• zero value, zero size, zero weight, zero time
• any value to zero? is it a value?
• integer 

Simmt Article Response

There is a quote at the beginning of this article that I slightly disagree with.  The quote states that "the study of mathematics is ultimately the study of humanity itself" (Devlin, 1988,9).  I would change the quote to say that mathematics is the study of nature, and not humanity.  I believe that mathematics is a man made construct to represent nature.  Mathematics helps us understand the world in which we live in.    
This article raised some valid points about mathematics education in schools contributing to citizenship.  Mathematics develops reasoning, patterning, critical thinking and generalizing.  I feel these qualities, without speaking in terms of mathematics, directly contribute to good citizenship education.  These qualities also are heightened with the skill of explanation, which I think mathematics helps develop.  Students practice communication in mathematics when they explain their though process to solve a problem.  Another aspect of math that develops citizenship education is the view that there is not a single right answer, but rather multiple ways to solve one problem.  This promotes the idea of diversity, different points of view and creativity.       

Fictional Letter from Future Students

Dear,
I wanted to let your know about a great math teacher I had in high school. She helped me see that math is easy and understandable.  Up until this grade I hated math and did not look forward to the semesters I had to take it. I probably began to dislike math when I wasn't doing well at it.  This teacher explained things in an understandable, simple and straightforward way.  She was able to bring this complex subject down to my level and not confuse us with details.  She was encouraging and caring because she often gave chances to redo some assessments to improve our mark.  She was very approachable and I felt comfortable admitting I didn't understand anything at all.  She was very patient and helped us see the answer for ourselves rather than just telling us how to do it.  Her lessons were fun and exciting because she didn't just talk at us.  We were all engaged and helping solve problems.  If I put my hand up to give an answer she never made me feel embarrassed if it was wrong.  
Regards.

Dear,
I wanted to let you know about a really bad math teacher I had in high school.  She confused me beyond belief and caused me to hate math for the rest of high school.  Her lessons were very confusing and all over the place.  She spoke to fast and didn't have time, or even know how, to properly answer student's questions.  The tests covered unexpected material which we hadn't become familiar with.  Her lessons were boring.  I would often space-out during her class because nothing was exciting.  She was often too busy to provide extra help and it was obvious she didn't put proper planning into her lessons.  
Regards.  

A few things I hope for is that I make my class fun and engaging where students want to get involved.  I don't necessarily want to only inspire my students in the area of math, but in life in general.  I want to be a positive role model for my students to believe in themselves and believe that they can do anything they want.  A few worries or fears that I have is that I will get stuck in a rut with my methods.  I worry that I will not be able to bring the subject matter down to their level.    

Battleground Schools: Mathematics Education - Article Response

While reading this article by Susan Gerofsky, I was surprised with the drastic changes which have taken place over the last one hundred years in the area of mathematics.  The Progressivist reform movement which took place around 1910 to 1940 was in response to the meaningless memorization procedures in the math curriculum during the late 19^th century.  The First World War brought about a need for people to be scientifically able.  Mathematics was brought to the front as an important subject area in the K-12 system.  This interesting because it seems that a major event needs to happen in order for the US to recognize they are lacking in a certain area to make a change. 

A similar situation also happened during the post-WW2 period when there was a fear that mathematics in the US was not keeping up with the rest of the world.  The response brought on The New Math during the 1960’s, and again seems to be a drastic change rather than slowly introducing and integrating new concepts and teaching methods into the system.  Adopting the polar opposite of what is already in place seems to never fully work.  Here, the government funded university mathematicians to write curricula, texts and teaching materials for the US and other nations.  It seems in this case, the drastic step was not well prepared. 

A simpler system was needed in response to the opposing debates and controversy over The New Math.  The Math Wars over the NCTM Standards from 1990’s to present time is the conservative response to the new progressive system.  The right-wing view is that teachers are experimenting and this is hindering the learning of their students.  I cannot help but wonder if a math curriculum can be developed which can include both a conservative and progressive approach.               

Assignment 1 Report

(Group Members: Michelle Davis, Hong Jiang and Nadine Lundie)


PART 1 – Mathematics Teacher

We chose to ask our five burning questions to two different math teachers with varying experience. The first teacher interviewed had graduated from the UBC four years ago and has been teaching grade 11 and 12 math. The second teacher is the Senior Math Expert and has 30 years of high school teaching experience. We decided to ask two teachers with different levels of teaching experience to compare our results.

We found the answers were quite different in that the responses from the recent UBC grad were responses that we as teacher candidates might have thought of. On the other hand, the responses from the experienced math teacher were quite different and her methods unexpected. For example, we asked both teachers how they approach both the students who like math and dislike math. The newer teacher said that when she has a student who is only there because it’s a requirement to graduate and they are not interested in pursuing anything mathematical after high school then she simply tells them what they need to pass. She explains the requirements for them to get a specific grade and lets the students decide for themselves what grade they want. The experienced math teacher did not bring up the notion of ‘requirement to graduate’ and said that she shows students that math is more than computation. She believes that math is a fine art and compares it to music in saying that playing scales is not all music is. She says that she teaches from the premise that math is creative, efficient, effective and fun.  She believes that all students can and should have an appreciation for mathematics even if they never plan to pursue it.  She is not content to just let them be.

This premise also relates to how she manages to relate math to concepts beyond the classroom. She describes math as a way of thinking which helps with organizational skills, efficient procedures and problem solving. She goes on to say that this mathematical way of thinking is used even for daily things such as your cell phone plan or relationship issues.  She firmly believes that math is more than just procedures to memorize and strives to make this come across to her students.

One thing our group found interesting was how the experienced teacher incorporated topics from the real world into the classroom compared to the newer teacher. Rather than presenting or telling about real world applications, the experienced teacher posed questions or problems for the students to think about. For example, using a parabola to represent a bridge or arch she would ask the students whether they could replicate a famous structure in a different location over another river of different width or to allow for taller boats. We found this very interesting because not only is she relating math to a real problem, she is also having the students try and solve it themselves. This would be a great idea for a group project. 

PART 2 – Mathematics Student

We asked our five burning questions to a high school student who likes math and a high school student who dislikes math. When considering math a mere subject, the student said that they like math because there is either a right answer or a wrong answer, and there is no maybe answers. The other student said they dislike math because they feel it is too hard for them to understand and they wait too long to ask for help until right before the unit test. Our group was wondering if perhaps the teacher was not approachable enough for extra help, or if the student already established a sense of defeat about math.

We found it interesting that the students had similar answers as to what about a specific math teacher made them their favourite. Both the students said that the teacher made math fun, used good humor and brought jokes about math into the lesson. For example, one of the students said that their teacher gave the three different forms for the equation of a line names that were funny and non-math related to help students remember. The student who disliked math said their favourite teacher explained things in the simplest and easiest way and the student who liked math said their favourite teacher did things that were hands on and interactive. One of the students also said their favourite teacher let them watch “Finding Nemo” during class.  It seems to us that it is much more about the relationship that the teacher built with the kids that made them memorable.

A few of the other comments we found interesting had little to do with the content of mathematics itself.  The student who likes math said that her teacher cared, wanted to help, and made sure that the students were doing okay, and not just in math. The student who dislikes math said that their favourite teacher had a positive attitude. We found this intriguing that these characteristics are completely unrelated to the subject. We have been learning that being a teacher is more than knowing and teaching your subject matter. It is also about caring for your students and their success.

Of course the issue of homework came up with both students.  They both said they did not much like homework, which is not surprising. The student who dislikes math said that if too many questions were assigned they would dread even getting started on it.  It’s an interesting issue; how much homework is too much?  How much is not enough?  As the teacher how do you treat homework in the evaluation process?

Conclusion

This interview was very eye opening for all the members in our group. We can relate to the answers from the newer math teacher, although we aspire to develop the methods of the experienced teacher. We believe the confidence and creativity this teacher brings to the classroom comes not only with experience but also with constant reflection and adaptation.

We gained a different perspective by interviewing both types of math students. We saw that they had different concerns in areas like what made learning the easiest, but they had the same ideas about homework and why a certain teacher were their favourite. We learned that the teacher having a positive attitude and caring about their personal wellbeing, which are both unrelated to math, are more important than we previously thought. 

Micro-Teaching Reflection

Every evaluation I received from piers said that they could identify all areas of the BOOPPPS lesson plan format.  I felt the hardest area to cover was the bridge, since we didn't know what would come right before my lesson.  I ended up relating my lesson to the one that finished right before and also gave a bit of background as to why I chose this topic.
All of the evaluations suggested different areas of development.  One was timing, because my teaching lesson didn't quite make it to 10 minutes.  I was expecting this suggestion because I felt that I rushed slightly through the lesson.  I had timed the lesson at home and it was 10 minutes but perhaps in front of a group the pace tends to speed up.  Another suggestion was to take into account that certain positions require flexibility that some students didn't have.  The third suggestion was to perhaps give individual feedback to each person while they are attempting the positions and give adjustments.  The last suggestion I recieved was to perhaps take the lesson further and teach ballet movements rather than just positions.  All of these suggestions were great because they were things I did not think of, especially giving individual adjustments and help.  Another improvement I would like to develop is incorporating more humor into lessons.  
All the evaluations highlighted that one of the lessons strengths was the post-test I did with candy as a prize.  I too was expecting this to be unique to my lesson.  
   

   

Micro-Teaching Lesson Plan

BRIDGE

WHAT-  Link my micro-lesson to the previous student's lesson.  Introducing what my lesson is about and why I chose it.  I took ballet many many years ago and this is one thing I have never forgotten.  To get started I will have everyone put on their "pretend tutus".

TIME-  2 minutes

MATERIALS-  Adequate space to spread out and our "pretend tutus"

LEARNING OBJECTIVES/OUTCOME

WHAT-  Students will know the five ballet positions and be able to demonstrate them.  

TIME-  n/a

MATERIALS-  a great micro-lesson.  

TEACHING OBJECTIVES/OUTCOME

WHAT-  The objective is to engage everyone and for them to see that ballet is easy!  I have the personal teaching objective of developing clear communication and easy to follow instructions.  I must watch that I don't talk too fast or go too quickly.

TIME-  n/a

MATERIALS-  The "pretend tutus" might help them relax and have fun which may bring them to want to participate more.

PRE-TEST

WHAT-  Has anyone ever been to see a ballet? and which ones?  Has anyone ever done ballet? Any other kind of dance?  Can you demonstrate a positions which comes to mind when you think about ballet?

TIME-  1 minute

MATERIALS-  none

PARTICIPATORY LEARNING

WHAT-  First, "ballet hands", soft elbows and posture.  Then I will go through the five positions first with them watching, and the second and third time through they will participate.  Lastly, once through from fifth to first position.    

TIME-  3 minutes

MATERIALS-  listening ears and watching eyes.

POST-TEST

WHAT-  Mini quiz: have students close their eyes, get into the position I say and have them open their eyes to check with other students.  

TIME-  1 minute

MATERIALS-  prize candy

SUMMARY/CLOSING

WHAT-  Give them ideas about how they can use their new skill- teach friends, show family, job interviews or even audition for a ballet company if you don't succeed at teaching.  And remember to take off your tutus and give them back to your ballet teacher!

TIME-  2 minutes

MATERIALS-  none

          

       

Dave Hewitt Video

I thought that Dave Hewitt's teaching methods were fantastic.  He developed the students mental math before he went into the visual aspect of it.  I think that he made the math less intimidating by saying it out loud and having them work it out in their head first.  I feel that if he would have written the equation on the board first and then explained the notation the students would have had a harder time connecting the it to the meaning.  I would definitely try to use these methods in my classroom.  I think the hardest part for me would be resisting the urge to say whether their response is correct or incorrect, because that's what the students are looking for.  In having the students figure out for themselves whether they have the right answer or not will help them develop confidence in their abilities.  

Remembering my Math Teachers...

In grade 7 I really began to enjoy math more than my other subjects.  My teacher was Mr. P and he was great.  I don't remember thinking it at the time, but I realized how good he was after I had another teacher for the next two years.  Mr. P's teaching method was very organized and simple.  He introduced a new concept with the minimum and didn't get carried away with details, applications or extensions of the idea.  Once the concept was introduced he would elaborate.  I remember him drawing a number line and moving along the line when he wanted to show addition and subtraction of negative numbers.  These visuals he used were very helpful to me.   What I learned from Mr. P is that in some areas of teaching math less is actually more. 

The next two years I had Mr. T and his methods were quite different from Mr. P.  He was great at math but had a difficult time explaining it in any other way than his own.  Many of my classmates were confused and he had a hard time clarifying their confusions and answering their questions.  I often helped my friends in class and I remember saying 'ignore all this and just focus on this...' because he presented so much material beyond the scope of the course that it confused many students.  He thought it would make more sense to students to go beyond the concept and introduce the extension thinking that would help the students to make sense of it.  But all it did was puzzle them because they were not  yet at that level of math to understand what he was even talking about. 

One more think about Mr. T, he must have mixed me up with another student on parent-teacher night because he told my parents all about how I was struggling and needed to do more homework to bring my mark up.  It was quite embarrassing. 

Reflections on In-Class Fraction Exercise

I thought that this activity displayed relational understanding vs. instrumental understanding perfectly.  By drawing circles and dividing them into groups or cutting each one up like a pie can really help students see what they are actually doing.  I am looking forward to finding ways to teach each topic like this.  

Assignment 1 Questions

Questions for the Math Student:


1.  What memorable method did a current or previous math teacher use that made the learning easier?
2.  What memorable method did a current or previous math teacher use that made the learning fun?
3.  Why is your favourite math teacher your favourite? What did they do differently than other math teachers?
4.  Why do you like (or dislike) math?
5.  How do you feel about the amount of math homework that you get?


Questions for the Math Teacher:


1. How do you incorporate topics into your math class which show how the math applied to real life outside the classroom?
2.  Do you include history of math into your classroom? How?
3.  What types of methods do you use to ensure that the student’s homework or assignments are
their own?
4.  What is the most challenging part about teaching the subject?
5.  How do you approach both the students who like math and students who dislike math?

Skemp Article Response

Through reading Skemp’s article Relational Understanding and Instrumental Understanding I have come to look at my days spent in math class in high school with a new light.    

                I often helped other students with their math and would be frustrated that the teacher would teach “beyond” or “above” the subject matter which confused the students who were struggling. However, I now see this would not have helped students when they encountered a slightly altered problem.  

               While I was in class much of the relational understanding concepts went right over my head.  I was only concerned with the instrumental understanding because that was all I needed to complete the assigned homework.  I usually understood the concept instrumentally at first, and then at a later time when I studied that material on my own I would come to understand it relationally.  I would see the larger picture and make sense of the material from class.

                I can see the teaching dilemma which is to teach for those students who have a difficult time relationally understanding the concepts.  It is tempting to resort to teaching for instrumental understanding to achieve higher examination scores.