Assignment 4: Unit Plan & Lesson Plans

 The following are my Unit and Lesson Plans on Polynomial Functions, grade 11 Math.

Polynomial Functions Rationale

Polynomial Functions Unit Plan

Polynomial Functions - Three Lesson Plans

Polynomial Functions Project

Foundations and Pre-Calculus 10 – Ski Hill Project

Learning Outcomes Addressed:

3.1 Determine the slope of a line segment by measuring or calculating the rise and run.

3.4 Explain why the slope of a line can be determined by using any two points on that line.

3.6 Draw a line, given its slope and a point on the line.

3.7 Determine another point on the line, given the slope and point on the line

3.9 Solve a contextual problem involving slope.

 Purpose:

The purpose of the project is to take what you have learned about slope and apply it to a real world scenario in which slopes are used.

 

Scenario:

An indoor ski hill recently opened in Dubai in the United Arab Emirates.  You and your two partners have been hired by a retail developer to propose a design for an indoor ski hill to be placed inside a mall in Southern California.  You will start with a short warm up to give you an idea of where to start; then you will be responsible for producing your own profile and a 3-D mock up of what the hill may look like in reality.

 Warm-up:

1)      Graph the following co-ordinates on grid paper. Join the points to create a simplified elevation profile of a ski hill.

Horizontal Position (ft)	Vertical Position (ft)
0	100
50	70
65	65
100	60
125	35
145	30
165	10
170	3
180	2
190	0
200	0

2)      Calculate the slope between the following pairs of points:

a)      (0, 100) to (50, 70)                                                                          m =                 

b)      (65, 65) to (100, 60)                                                                       m =                 

c)      (145, 30) to (165, 10)                                                                     m =                 

3)      Which part of the run has the steepest slope?                                                                

4)      Find the slope for the last 10 feet of the run.                                           m =                 

5)      Find the average slope from beginning to end.                                        m =                 

6)      Calculate the length of this ski run.                                                          l =                   

Your Proposal:

Your proposal is to include three aspects.

1. Your ski hill elevation profile is to meet the following specifications:

·         The total height of the ski hill should be 100 feet.

·         The total horizontal distance should be 120 feet.

·         The total length of the run should be no more than 300 feet.

·         You should include at least 3 different slopes and a more gentle finish to allow the skier to slow down and stop.

·         Your drawing should be on grid paper with an appropriate scale given.

2. You must also include a paragraph description of the ski hill with the following information:

·         The total length of the ski run.

·         The slope of the steepest section.

·         The average slope of the ski run.

3. Your 3-D Model is to meet the following specifications:

·         Must be done to scale, ideally the scale you used for your elevation profile.

·         Must include your scale on your model.

·         Must be able to see the 3 different slope sections and the gentle finish.

·         Must look like it would in reality, i.e. make it look like you would see it in the mall (add snow, the lift, skiers, etc…, be creative)

Timeline:

·         You will have one class period to complete the warm up and discuss what your proposal will look like.

·         You will have an additional 2 class periods to work as a group to develop your profile, written description and model.

·         If additional time is needed the classroom will be open after school for one additional week so you may complete your work at that time if necessary.

Extensions:

If you wish to go above and beyond the project specifications here are some ideas for you; you may choose one or you may choose to do them all.

• Research the Dubai project – cost to build, timeline from start to finish, slopes involved, cost to keep hill operational.
• Research the practicality (cost to build, marketing, cost to run yearly, etc…) of developing an indoor ski hill in a Western Canadian mall.
• Research how outdoor ski hills are run and how they may use slopes to classify the different runs.
• What is the connection between man-made ski hills (not necessarily indoor) and polygons?

 Rubric for the ski hill project:

Knowledge and Understanding - Calculations show complete understanding of the mathematical concepts used to complete the assignment. All calculations are done correctly; co-ordinates are graphed accurately on grid paper.         /10

Completeness of Assignment - Contains all the required parts: graph, slopes, steepest slope, slope for the last 10 feet, average slope and total length of the ski run. The proposal for the ski hill elevation profile meets all required specifications. Includes a brief description with all required information .     /15                                           
Neatness and organization - The work is presented in a neat, clear, organized fashion.      /5 
                                                                                              Total      /30

Review of a math 9 polyhedra project created by Susan Gerofsky Experience:

With Deborah and Nadine

Researching the four topics was very interesting although obtaining information for part one was difficult. Also the mathematics involved for some parts is too complicated for a grade nine to understand. No guidance as to the weighting of parts a, b, and c is given which could lead to some groups putting emphasis on the biography and others on the mathematical ideas. Making the origami model was fun for those of us who enjoy art projects; however this task is frustrating for those who do not.

Overall we found this project interesting but dated. There was no relationship with today’s world and we question the value this project has beyond providing interesting facts.

Benefits:

* Students will learn interesting facts through research and from peers rather than their teacher
* History is incorporated into the classroom, enriching the learning experience
* Students will see how mathematical ideas can extend into the arts and architecture, making mathematics seem less isolated as a subject
* Learning about mathematicians, connecting math to real people can spark interest and provide comfort to those who may see the subject as alien
* Allowing students to create and thus touch and experience in real life the Platonic solids allows them to realise their simplicity


Weaknesses:

* It is unclear as to why some people work with partners and some alone to write the paper
* There are no specifics on whether each person writes about a different part or all contribute to every part, writing an essay with someone else is not easy
* Each person has to make an origami model, time consuming & some students may struggle with this, not beneficial, there needs to be more of a point to making them
* Other than introducing polyhedra and some history the intended learning outcomes of this project are not obvious
* No specifics for what the presentation should look like
* Students may not connect with mathematicians from the past
* The math involved in some sections is too complex for the grade level
* No connections to the lives of the students
* Group work, especially of this nature, is very time consuming



Uses:

* Introducing the class to polyhedra, good start to 3D geometry
* Can be incorporated into volume, or surface area units
* Allowing the class to experience group work and obtain skills in researching
* Incorporating history into the math classroom



Modifications & Extensions:

* Only one student per group makes an origami model, the teacher can bring in the fifth one. They show the template to the other group members before gluing so that all students can see how it is constructed and benefit
* All students should be involved in the research, then two can work together to produce the written work, one can present and one can make the model so that everyone contributes, if they want to change that they can but they must inform the teacher of their roles
* Find modern day uses for polyhedral, compare with historical
* Research more than one of the areas or people then groups can compare
* Presentation only, no essay
* Using paper templates students can cut out all kinds of regular polygons with the same side length, then try to create all kinds of regular 3D objects by gluing the regular polygons together
* Find patterns in the construction of the solids
* Use Geomag for constructing the solids (stick magnets with iron balls, easier and more fun to use than paper and glue)

Response to "Creativity, flexibility, adaptivity, and strategy use in mathematics"

Creativity, flexibility and adaptability are all necessary attributes of problem solving. I think the aim of teaching mathematics should be creative problem solving, to encourage students to seek out solutions, exploring patterns, finding new strategies- not just memorizing procedures, formulas and doing exercises. In order to achieve creative thinking in the classroom, students should be encouraged to use different problem solving strategies.

This article ties in well with what we have explored earlier this term about relational and instrumental learning. Instrumental learning provides a narrow curriculum based on mastering facts and procedures which does not allow students to become creative thinkers. Relational understanding, on the other hand, trains students to think creatively and develops their problem solving ability. Mathematically powerful students are capable of interpreting large amount of data and this allows them the flexibility and adaptability to come up with new strategies or to choose the most appropriate strategy to solve a problem.

Ferit's compensation strategy, adding 100 first and then subtracting 1, worked for 127+99, but he did not get the right answer for numbers such 133. His creativity enabled him to come up with a strategy that was new to him or at least unfamiliar to him. The lack of a solid mathematical background prevented him to switch between different strategies, he used the compensation strategy for all of the addition problems he encountered getting the wrong answer most of the time.

Two Column Problem Solving

The Restless Fly from "The Moscow Puzzles" by Boris Kordemsky

Two cyclists began a training run simultaneously, one starting from Moscow, the other from Simferopol.
When the riders were 180 miles apart, a fly took an interest. Starting on one cyclist's shoulder, the fly flew ahead to meet the other cyclist. On reaching the latter, the fly at once turned back.
The restless fly continued to shuttle back and forth until the pair met; then it settled on the nose of one of the cyclists.
The fly's speed was 30 miles per hour. Each cyclist's speed was 15 miles per hour.
How many miles did the fly travel?

The problem looks more complicated that it actually is. At first, students might say there is not enough information given, since we don't know the distance between the two cities. On the other hand, although it is highly unlikely that a fly would travel from biker to biker until the two meet, students will probably find it funny and will try to figure out the solution. It is not a practical problem, but probably it is both memorable and strange because of the travelling fly. I still remember some physics problems where we had to determine the speed and angle of a brick thrown out the window of a tall building so it would land on the pedestrian's head who was walking by. It was a weird problem but interestig enough for us to figure out the solution to it.
We could extend the problem by having different speeds for the cyclists and maybe having another fly travelling from the opposite direction with a different speed.

If the two cyclists had the same speed, they meet halfway, both travelling 90 miles. Since the fly's speed was twice as much as the cyclists' speed, and flew without stopping, it had to travel twice as much, a total of 180 miles.
If we look at the time spent until the cyclists meet, t=90/15, we get 6 hours. The fly travels between the two cyclists until they meet, for the same amount of time. This means that the distance covered by the fly will be d=30x6, d=180 miles.

Short Practicum Stories

#1

My short practicum started with an unexpectedly warm welcome from the school administration followed by the tour of the scool. We (together with two other UBC teacher candidates) were introduced to almost everybody in the school and spent about an hour getting to know the school policies and procedures in the board office with all three vice principals and principal in attendance. I really appreaciate their effort to make our short practicum a positive experience. All the teachers of the Math department heard about me coming to observe and teach in my SA's class. Two of them came over to see me teach when they had a free block. I got a lot of advice from so many of them, I fell really thankful.

#2

I wasn't aware that the day my FA was coming is actually going to be a modified day, so I prepared a nice long lesson on the quadratic formula and the proof of the formula. I had to adjust the lesson to be able to finish it in 50 min insted of 70 min but wasn't able to go over everyting I have planned.
During the lesson one of the student's desk he was sitting on fell apart and the whole class laughed for about 5 minutes, which shortened my already short time. At almost the end of the period the announcement went on eating another 5 minutes not leaving me time to wrap up the lesson. It was really frustrating but I learned that I have to be more flexible as a teacher because situations like these happen and I have to be able to deal with the unexpected all the time.

LESSON PLAN ON AREAS OF RECTANGLES,CIRCLES AND SQUARES with Feda and Namrat

	What	How long	Materials used
Bridge	Review the definitions of rectangles, squares and circles. Take several responses to get an idea of what the students know.	2 min
Learning Objective	Students will calculate the area of a rectangle, a square and a circle. Students will solve problems involving the areas of rectangles, squares or circles.
Teaching Objective	Students will be able to solve problems individually and cooperatively. Students will be able to clearly and logically communicate a solution to a problem and the process used to solve it.
Pre-test	Ask the students about the formula of areas of rectangles, squares and circles.	2 min
Participatory learning	Ask the Students to solve a problem that involves the calculations of the area of a rectangle and a circle combined. Ask students to describe in words the steps that they will need to take to obtain the answer. Students will work in small groups to find the areas by using the formulas for the area of a rectangle and of a circle.	5 min	Rectangle shaped cardboards with a cut-out circle in the middle.
Post-test	Ask students to solve a similar problem involving many circles marked on a rectangle. Compare results with other groups.	5 min	Another cardboard with many circles marked on a rectangle
Summary	Give overview	1 min

Second Microteching (with Feda and Namrat)

Feedback from students:

The majority of comments on our microteching were very positive, most students thought that our lesson was well structured and paced and they didn't have difficulty following through the steps. They found the lesson engaging and liked the participatory activity. One student mentioned that our cardboard rectangles were more useful during problem solving than a diagram on the white board. A few students commented that the intoduction of the lesson was not grade appropriate, it was too simple for grade nines. One student commented that we have to pay attention to the whole group when explaining a concept and make sure that students at the back can hear it too. One student didn't find our example to real life applications
very relevant.

My conclusion of the lesson:

I think our lesson was well planned and structured, and most of the feedback we got was very positive. The hardest part for me was the lesson plan, and to think of an activity that is both engaging and age appropriate. Now that I had my first teaching experience in high school, I can disagree with the students who thought that the lesson was too easy for grade nines. I found the feedback very valuable since it gave me the opportunity to think of the areas of my teching that need to be improved. I liked both the instruction part and working with a small group. Comments regarding connections with real life applications made me think of other ways of enagaging students in learning activities. If they see a reason behind the concept, students will be more motivated to learn it.

Thinking Mathematically (Chapters 2 and 3)

For most of us, whose brains are mathematically wired, it seems obvious and natural to break down any problem solving question to the three phases: Entry, Attack, and Review. Most students, on the other hand, are rushing to figure out the solution right away, without even getting to the end of the question. And this is a problem because not reading the problem careful enough, they lose important information about it. By rereading the problem, and asking the question: “What do I KNOW?”, the students will be able to gather important information and find better approaches to solve it.
I think this first phase is crucial and more important than the other two, because without fully understanding the problem students won’t be able to get to the solution therefore it is worth spending extra time with it.
The second phase, the Attack, (What do I WANT?) has its importance too: you cannot solve a problem if you don't know what do you want to get out of the problem and soon might find yourself in a totally different direction.
The Reflection phase, ( What can I INTRODUCE?) is an opportunity for students to come up with different ways of solving a problem and maybe adding to it in complexity.

Poem About Zero

Most say that I am a Zero, a Nothing,
But think about it my friend,
The less you have of me the poorer you are
The more the richer.

I’m not proud of the fact
That bankers use and misuse me,
And, the debt they created
Is measured by a long sequence of me.

It is them, bankers, who
Divide and conquer,
Not mathematicians who say
That dividing by zero is infinity.

Timed Writing

Divide
there are language barriers that divide new immigrants from the native population
divide and conquer
share, divide equally,
share thoughts
mathematics, basic numeracy skill, i can' think of anythink else right now
bankers don't divide equally with the rest of us, dividing with zero doesn't make sense, the earth is divided into northern and souther hemispheres, division is the opposite of multiplication, negative divided by negative is positive, negative divided by positive is negative and vice versa,

Zero

Nothing, emptiness, NIL, hollow, number, word, group of letters, infinity, black hole, space, you cannot divide with zero; some have zero, some have all; for some zero is not a number; in mathematics it has its rightful space, zero money is no money, Mathematicians think that dividing with zero gives you infinity, other people say that dividing with zero is not allowed, disallowed value,zero is the middle of the number line, zero degree celsius is freezing cold, zero divided by any number is zero, there is no negative or positive zero- zero is signless,

Response to the "Citizenship Education in the Context of School Mathematics" article

I totally agree with Elaine Simmt, that mathematics education is crucial in the development of informed, active and critical citizens in today's society.
The study of mathematics cannot be resumed to the simple aquisition of knowledge, but it must focus on developing a way of thinking trough a permanent training of the mind. Thus, learning mathematics results in developing capacities that are necessary and useful in the every day practical life.

Learning mathematics results in logical thinking and reasoning, it creates the capacity of the individual to analyze and make well-thought decisions. Mathematics is an exercise of the mind, it is a preparation for all challenges of life, profession and career. Therefore, mathematical knowledge is essential and crucial in order for an individual to be active and critical participant in our society.

It is very important that us, future teachers, make our students love mathematics just as much they love their toys, bikes, computers or video games. By teaching them how to decipher the secrets of mathematics they can reach limits which, otherwise would seem unattainable. Our role is to convince our students that by learning mathematics they can have a better understanding of the world and they have the power to shape their future. They have the choice of becoming critical thinkers and decide for themselves rather than becoming frustrated, obedient and thoughtless consumers.

Hopes, worries and fears for the future

I am hoping that my choice of becoming a teacher was a good one and I will be able to touch my students' lives in some way or other. I would like to be there when my students need any kind of support and let them know that they are important and their opinion counts. Also, I would like to change their misconception that math is hard, and make them believe that everybody can learn it if they put their effort and mind into it. I realize that teachers can have a great influence on how their students view themselves and the world around them, and I would like to help them become critical thinkers. My fear is that being a perfectionist, I will push my students too hard and instead of liking math they might start hating it.

Letters from former math students

Letter #1
Dear Mrs. S,

It' been 10 years since our high shool graduation, when you had your farewell speech in front of our class. At that time I didn't care much about the world around me, but your words touched me so much I can still remember them. I did chose a path in my life I thought would suit me, and you were right, you have to love what you are doing or else you wouldn't be fulfilled in your life. Although I struggled with math you thaught me to think for myself, to see the world with my own eyes and become a critical thinker. I do appreciate that you cared enough for me to not let me quit school when I felt I can't go on anymore.
Thank you for caring,

William

Letter #2
Dear Mrs. S,

Ever since you told me that no matter how gifted I am in math I have to work hard in order to get somewhere I was trying to prove you wrong. Unfortunately this didn't get me far in life and I am blaming you for it. If it wasn't you I would have succeded a long time ago and would have enjojed life like everybody else.
Probably you are an even bigger expert in torturing students, since you have 10 more years in doing it so. I feel sorry for all of your students, because I believe that life should be enjoyed to the fullest when you are a teenager, and your expectations do not allow for that. School should be fun and this would be only possible without homework.
I hope you retire soon or find some other job for yourself!

Mark

Summary of “Battleground Schools: Mathematics Education” by Susan Gerofsky

This paper is discussing the ongoing and never ending battle around North American mathematics education and the reform movements of the 20th century. The three reform movements: the Progressivist (1910-1940), the New Math (1960's), and the NCTM Standards-based Math Wars
can be seen as battles between conservatives and progressivists. Since the existence of public schools in North America, in the late 19th century, there were many public criticism of school mathematics as a process of meaningless memorized procedures without knowing why these particular procedures worked.

The pressure for more meaningful mathematics curriculum increased after the First World War with the increase in international immigration and the rapid changes in society.
John Dewey's work is particularly important because he challenged the Cartesian split between knowing and doing, or abstract and applied knowledge. He believed that students thrive in an environment where they are allowed to experience and interact with the curriculum, and all students should have the opportunity to take part in their own learning. The role of the teacher should be the facilitator where learning situations and materials are carefully structured and prepared in advance. Dewey said that an educator must take into account the unique differences between each student. Thus, teaching and curriculum must be designed in ways that allow for such individual differences. Although Dewey's ideas won a high degree of acceptance in progressive teacher's colleges, most North American classrooms followed a very conservative approach.

After the Second World War both educators and the public recognized the need for more technical and mathematical skills. After the surprise launch of the Soviet Sputnik satellite in 1957, improving mathematics education at the K-12 level became of utmost importance. The resulting movement was called “The New Math”, which gained momentum in 1960 and its influence spread worldwide. The New Math supporters were highly conservative except for a few progressive ideals: they supported understanding over fluency, and to some extent, inquiry and sense-making over absorbing and applying facts.

The NCTM Standards were shaped by both constructivist and progressive approaches emphasizing the development of flexible problem-solving skills, the ability to represent mathematical relationships in multiple forms. The use of calculators and computers was encouraged as an essential part of the problem-solving process. Students should also be encouraged to devise their own plans and explore alternate approaches to problems to gain the ability to communicate mathematically.

The Math Wars today are far from over yet and I think it is very important to separate the education system from political interests. Teaching mathematics should not be dictated by economical or political movements, the goal of teaching and curriculum should be designed to fit the needs of the students regardless of the socio economical circumstances. It would benefit the society as a whole if we could engage students in a reflective inquiry thus increasing their intelligence and knowledge which can be applied to all areas of life.

Microteaching Self-Evaluation

Most of the students in my group were excited about my lesson topic, since they've never tasted a green smoothie before. All three of them agreed that it was very informative and they learnt a lot they hadn't known before. They found that the introduction/bridge was clear, one of them noted that the lesson could have started with listing the health benefits of the smoothie first. They all liked the paricipatory hands-on activity,(they touched and tasted the greens). One of the students noted that there was a bit too much lecturing. All three of them found that my time management needs improvement since the smoothie had to be made in a rush at the end of the lesson. All three students were excited about tasting and drinking the smoothie, one of them noted that she wants to try the recipes at home.

Reflecting back on the lesson, I think I used too much time at the beginning lecturing, this is why I ran out of time at the end. Also, I was a little anxious troughout the lesson, probably this could have been prevented if I would have practiced it beforehand. I liked how excited my students were learning about the health benefits of green smoothies. I think that showing students the different kinds of greens, then asking them to touch and taste them was a good addition to the lesson. I also think that the handout I prepared with the health benefits was well appreciated.
For the future I really need to pay attention to timing to avoid losing important
parts at the end of the lesson.

Student and Teacher Interview Summary

By: Raman Dhiman, Zsofia Szigeti, Marija O’Neill


We interviewed a teacher with more than twenty years of experience both in public and private schools. She is currently teaching grades nine and ten. Following are some of the highlights of our interview including interesting points from teacher’s answers and our responses.

When we asked the teacher how she manages students with different abilities and work habits she said that she does not like to give her advanced students work which is ahead of the curriculum. Instead, she keeps them busy by giving them work that is broad in the subject and by encouraging them to help others. As a parent, I would like to have this teacher teach my kids, although this may not be a wish shared by some other parents. There are parents and kids who focus on raised goals, raising them as far ahead of curriculum as possible. I like my kids and my students to enjoy more stable growth and not be working ahead for one period then be bored the next, and then possibly even loose the academic momentum, work habits, and the ability to look into the subject deeper.

We were impressed that she was fond of using technology such as Tablet pc, overhead projector, and the Internet. The tablet pc appears quite helpful to give live displays of the graphical presentation, while she is facing the class.

When we asked her what she finds most rewarding about being a teacher she responded by saying that creating a safe environment where students feel understood and that they matter was on top of her list. She added that she enjoys the fact that she can focus and direct their attention to certain things without controlling them. Students on the other hand, are active participants in the learning process because of teacher’s ability to keep them interested in learning.

Showing genuine interest in students and listening helps determine their needs, which allows a teacher to adjust the curriculum accordingly. Offering help after school is very important because there are students who are shy and do not dare to ask questions in class. Being too shy to ask questions in class is not something I had ever previously given much thought about. Given that many students find math difficult she takes extra steps in motivating and even has quotations on her wall such as “It is the attitude not the aptitude that determines the altitude of you success” and “I-m-possible”.

Being a substitute teacher appears to be the most difficult position to be in because there was no relationship between the teacher and the students and she could not bring her own material to make her lesson more appealing. We may be in this position before we get a permanent job and it is good to know that.

Student's interview:

In addition, we interviewed a grade ten student whose strengths she stated are not in math. She shared her thoughts about her learning experiences as well as her emotions towards the subject of mathematics.

In the very beginning of our conversation with the student we gathered that she was one of those typical kids with fear of math. She said that math was the hardest subject and feels very nervous about it. Even if she knew that she is using the right methods to solve problems she was never fully confident about the outcome. The anxiety would usually be over once she had a confirmation that her answer was correct. As a result, she needed more time to complete her work in class.

When we asked what she liked about her teacher she said it was the fact that her teacher listens to her students and adjusts curriculum based on what she feels suits them. For example, the teacher allows them to take extra time to complete their exercises in class if necessary. One of the things she did not like about her teacher is that although she would give enough time to finish work in the classroom, she did not allow enough time to prepare for exams. Also, she would like her teacher to help her build confidence that she needs in order to tackle some of her mathematical challenges. She realizes the importance of math in her daily life. Some of the examples of math applications in her life were: addition and subtraction, percentages in the stores (for discounts), and interestingly she mentioned speed again.

One of the greatest discoveries about her learning was when we asked if she would like to have a career related to math. She answered that she was fascinated with proofs and how and why mathematical theories work. She is one of those kids who takes time to think what the meaning behind a quadratic equation, for example, is and not just trying to solve it quickly. This is why she is slow and needs extra time. From this article, we learnt that we as future educators must be aware of HOW students learn and maybe investigate the reasons why some students take longer to perform math operations.

Thoughts on David Hewitt's Teaching

 
David Hewitt’s teaching style is interesting and impressive at the same time to me.  I’ve never seen anybody teaching this way before. His auditory teaching style without the use of the visual made me realize that there are many possible ways of teaching that can work better than the ones I knew before. I am not sure if someone can learn to teach this way or it’s just a special skill but it is definitely a very effective method of teaching. The loud banging noise he makes with his stick gets all  his student’s attention and it seemed that the students were following along. It was interesting for me to see him do the transition from the number line to equations.
 
 

How to make a smoothie

Name: Zsofia Szigeti	Lesson Plan	Date:22 Sept 2010
What	How Long	Materials
Bridge: Introduce the concept of green smoothie to the students. Ask students to list some of the green leafy vegetables they know.	1 min
Learning Objectives: Students will be able to identify green leafy vegetables and make their own smoothie.
Teaching Objectives: Students learn about the nutritional value and health benefits of dark green leafy vegetables.
Pretest: Ask students to identify the different types of green leafy vegetables.	1 min	Kale, Russian Kale, Swiss Chard, Red Chard.
Participatory Learning: Students will explore different smoothie recipes and develop their own.	4 min	Print-out with smoothie recipes and health benefits.
Post -Test: Students choose a recipe and make their own smoothie.	3 min	Blender, Fruits, Greens, Water.
Summary and Closing: Ask the students about the health benefits of greens in our diet.	1min

My Memorable Math Teacher

   The amount of math we did in high school was so huge that it didn't leave time for social interaction between us, students, and our math teacher. We hated the extra amount of work he gave us because even without that extra work we spent 2-3 hours studying math.
Sometimes, instead of coming to school at 8'clock he asked us to come an hour before, at 7'clock, so we can better prepare for our university admission exams. During winter it was almost dark at that hour and we hated getting up at 6 for an extra hour of math and did not appreciate his efforts. He wasn't paid for that extra hour but cared enough for us to sacrifice one hour of his sleep to make sure that we know everything we need to make it to university. At the end of grade 12 we had our exams for admission to different universities in the country and 30 out of 33 made it that year, the rest of 3 the following year.
  The most memorable thing about my teacher though has not much to do with math. It was our one and only two day class bus trip, going halfway across the country to visit the hometown of the famous Hungarian mathematician, Janos Bolyai. It was late November, dark and raining and our bus broke down. All of us had to get off the bus so the driver could be able to fix the problem. We were told that it will take at least an hour to fix the bus so our teacher decided that in order to keep ourselves warm we have to start walking until the bus will catch us up. We were cold and wet and our teacher's gesture to offer us his scarf, his hat and his umbrella made us realize that he was more than a teacher to us. He taught us because he cared for us and I carry his memory in my heart forever.

Questions for a High School Mathematics Teacher and a Student

(Raman, Maria and Zsofia)
Questions for a teacher:

1. What do yo do when advanced students have finished their work in class well ahead of the rest of the class, do you give them extra work?
2. What kind of technology do you use?
3. What do you find most rewarding being a teacher?
4. What challenges did you face as a new teacher?
5. What teaching strategies work best with your students?

Questions for a student:

1. How do you feel about solving Math problems?
2. What do you like about your Math teacher?
3. What do you not like about your teacher?
4. Do you find it useful in your daily life?
5. Would you like to have a career related to Mathematics?

Response to the Richard Skemp article

Instrumental Understanding vs. Relational Understanding

It is my experience as a tutor, that most students prefer instrumental understanding versus relational understanding and most math teachers are in favor of the former. Instead of trying to solve a given problem, students with instrumental understanding are focused on getting the solution right, rather than using logical reasoning. I have seen students do the calculations of the values of a function perfectly but getting stuck at graphing the function because they don't see the biger picture. Many of these students have an amazing speed and accuracy at computation but are lacking in understanding even the basic principles of Mathematics. This ability at computation gives them an advantage over students who use relational understanding, students who have a deeper understanding of principles but are lacking in computational skills. These students know how to figure out the solution to difficult “mindbenders”, but, often make errors and are a lot slower in performing computations.

I find it unfortunate that most students in our school systems are taught instrumentally. This might be due to the structuring of a one year curriculum into one semester, this way teachers have to rush trough material making sure they finish it by the end of the semester. It is hard to accommodate students whose goal is to understand relationally since the majority of each classroom are students who understand instrumentally. Most teachers also prefer students with computational abilities and are not willing to change their teaching style in order to accomodate relational understanding. Even most regional and provincial level individual Mathematics competitions are designed to fit the learning style of those who understand instrumentally. In team competitions though, where higher level critical thinking is required, students with relational understanding thrive, they have a great advantage over the former group of students.