Notables 2: "Population, Arithmetic, and Energy" by Prof. Albert A. Bartlett

Population, Arithmetic, and Energy by Prof. Albert A. Bartlett

This is an 8-part presentation by Prof. Albert A. Bartlet on the importance of understanding the exponential function and growth rate.

Assignment #3: Doing and Assessing a Math Project and Designing a Math Project

https://docs.google.com/fileview?id=0B01xnh0Ekjr6MzkxNmNjYzgtNWU5OC00NzU1LTg2YzItZmEyNmRmM2VmMmU3&hl=en



https://docs.google.com/fileview?id=0B01xnh0Ekjr6MTY0YmQ0ZDQtZGM0OC00NTU0LWExOGEtNGI4MDFkOWMxZjY3&hl=en

Two Column Problem Solving

Cycling Digits from Thinking Mathematically (page 165)

Math Text Book Assessment

A group of us examined the Math 12 text book by Addison Wesley
Group members: Amelia, Mina, Rory, Rosemary, and Sam

• 7/10 for heaviness. Quite solid, but not too heavy. Uniformly dense.

• Durable enough to be thrown on the ground.

• It smells like teen spirit.

• Pages are typical with regard to thickness.

• Latest publication date = 1999.

• Expected durability = 5-10 years.

• Authors: Robert Alexander & Brendan Kelly + 11 co-authors.

• Authors are mostly teachers and math educators.

• Preface lays out the general format to be used in the upcoming chapters.

• Chapters are based on specific units, which are further subdivided into different topics.

• Typical answer key, glossary and index.

• Rory was pleased by the frog on the front cover.

• Page layout uses a good amount of white space and doesn’t overwhelm the reader with information.

• Good use of photos and diagrams. Varied fonts for overall aesthetic appeal.

• Friendly and accessible.

• PLOs for Trig in the IRP have more emphasis on unit circles than in AW.

• Extra material on probability, stats and conic sections.

• Chapter sequence is logical. Builds up concept of function and expands with sinusoidal & logarithmic functions.

• Includes material which correlates to PLOs; ie: citizenship education (equality of sexes, etc.).

• Incorporates use of technology.

• Usefulness for new/experienced teachers?

• Usefulness for students/self directed learning?

Notables 1: 2009 Benjamin Franklin Medal Winner: Lotfi A. Zadeh for Fuzzy Logic

Fuzzy Logic, Lotfi A. Zadeh

In-class timed writing (8 min) - Two memorable stories about the practicum experience

The "AH... I get it!" moment

It happened when I was teaching a math 11 class. I was explaining the standard form of a the most general quadratic function and its graphical representation as a parabola and having a dialogue with the students through looking at equations, having a table of values, and graphing. I demonstrated how different parameters in the equation were responsible in changing the shape of the graph, translating it along the x- or y-axis, and reflecting it about the x-axis. We were doing few examples, going back and forth between the table of values, the equation, and the graph, determining the axis of symmetry, the vertex, the opening direction, the x- and y-intercepts, ... It was at this point where one of the students from the back row yelled with excitement "AH... I get it!". At that moment, I stopped, acknowledged him, and told the whole class, "I love these "Ah ha" moments!


The "Jigsaw activity" for matching compound names and formulas
My SA suggested a jigsaw activity for helping the grade 9 science students in naming and finding chemical names for chemical compounds. I took the 2nd half of a science 9 class to the gym, handed out pieces of cut paper with names either the name or formula of a compound, and asked them to go around the room and find the other student with the corresponding formula or name. Once they found their partner, they were supposed to sit down. At the end, when everyone had found their partners, I went around, picked few students and asked them how they found the matching name or formula. I felt like I had a strong presence and had convinced the students of the significant point of the activity, i.e. not memorizing how many electrons a given element is willing to give or take to form an ionic bond, but to know how the exchange is made and how many atoms of each element were required such that there is a balance of charges at the end, forming a electrically neutral object.

Poem on "Division by Zero"

Constraints: At least 8 lines long, Can't rhyme.

How many zeros can you fit in anything, any number, any shape, any place, at any given time?
Are there infinite zeros in anything, any number, any shape, any place, at any given time?
What is revealed when diving by zero anyways? What is infinity?
Does a task or a chunk become smaller when it's divided by zero?
Division by zero looks like something on top of zero.
There is a line Between the number and zero when dividing that number by zero.
Divide and conquer does not work when dividing by zero, does it?
Division by zero does not imply inverse of multiplying by zero!

Reflection on this exercise:
A worthy exercise, but I don't see/get its relevance or importance.
Maybe because my background is in physics?

Would you use it in your classes? How?
I don't know... Honestly, probably not. I'd rather use something or some
activity related to sports than poetry when discussing a concept. This
is because I have participated in different types of sports all my life.

In class timed free writing on Zero and Division

We were asked to do two consecutive in class "free" writings, each 4 min. long in duration. We were supposed to write continuously, writing "I can't think of something to write" repeatedly until we had a new idea/sentence to write. In transferring my writing from class, I have marked those instances by replacing the above statement with "..." for brevity.


Timed "free" writing on Division (4 min):

Making something big small. Making distinctions. Individualizing or categorizing. Dividing a room. ... Making tasks simple. ... Breaking to smaller chunks. What does it mean to divide a number by itself? What does it mean to divide a number by zero? What about dividing zero by any number? Dividing zero by zero? Making a long journey into smaller segments. Making shapes into smaller parts.


Times "free" writing on Zero (4 min):
Looks like a circle. Is the first integer, the middle point between negative and positive numbers. It is elusive! How many zeros can you fit in a zero? How many zeros can you fit in one? What is the size of zero? Why do you get zero when you multiply another number by zero? You get the same number when you add it to zero. A number minus itself is zero. ... Zero is complex, yet so simple. Its whole, yet empty. Can God be zero? ... Zero is in the shape of a circle. 0! (zero factorial) is one, why? ... Zero means focus: zeroing on things, narrowing down. Zero is nothing, nil, nothing, zilch, notta.

Assignment #2b: Group microteaching feedback and reflection

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 humor, 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.


Reflections
I felt a bit of planning and practicing with the group prior to our teaching would have proven instrumental. Just as any group activity/project, communication, planning, coordination, and practice are essential. We did not get a chance to do this because of scheduling conflicts and due to the sheer amount of work for other classes.

I liked the interaction and question/answer session with the students during the review session. This allows for the teacher to touch basis with the students and find out what and how much they remember from the last class/topic (arithmetic sequences/series) before moving on to introducing the new topic (geometric sequences/series in this case).

Developing informed, active, and critical citizens in my math classroom

Reflections and ideas based on reading
CITIZENSHIP EDUCATION IN THE CONTEXT OF SCHOOL MATHEMATICS"
by Elain Simmt

Undoubtedly, mathematics plays an important role both in terms of numeracy and logic in not just today's society but throughout ages and in varied and diverse societies all over the world. Once sees applications of mathematics in nature, art, science, technology, business, economics, sociology, health, meteorology, sports, politics, architecture, transportation, spirituality, ... This is in form of numbers & figures in quantitative measurements, shapes and geometry, probability and statistics, and diagrams and graphical representations. Examples of applications range from describing natural phenomenon through mathematical modeling and devising multi-parameter models to predict reality to using power law to explain growth/decline rates and fractal geometry for image compression, Mathematics also forms the basis for decision- and policy-making, economic and financial transactions. We even use mathematics in our day-to-day lingo: weighting one's options, trends,

Mathematical thinking, through the processes of posing questions, discussing and devising different approaches to solving problems, using critical reasoning to making sense of conceptions, analyzing current events which quote or use mathematical results, recognizing patterns, making connections, making estimations and educated guesses, communicating and describing results, offering explanations, ..., enable and empower students to apply these tools in other aspects of life and outside the classroom.

I believe individuals should not be accepting the so-called "facts" at their face value. They should instead be constantly asking questions like: "does it make sense? Is this right? What does this mean? What is this trying to tell me?, ..." I intend to instill this belief in the students I cross paths with.

Assignment #2a: Group microteaching lesson plan

Topic: Geometric Series
Time duration: 15 minutes
Group members: Mina, Rory, Sam

Bridge/Pretest: Recall arithmetic sequences and series, and 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 ric 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 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.

"Art of problem posing" by Stephen Brown & Marion Walter

How would you use "Art of Problem Posing" in micro-teaching?

By composing and posing a series of questions on a new topic, one is able to introduce the concept without "telling" what the concept is. In a sense, it becomes a process of search and discovery; exploration and illumination and therefore more meaningful. Learning and teaching becomes more collaborative/cooperative.

What do you think of the "what-if-not" approach?

The "what-if-not" approach stresses how things should not be taken at face value. This type of questioning opens doors to other paths of solving/analyzing problems and another way of thinking about them. In a sense, there is more generalization of a concept going on. More in terms of relating the problem/issue to other similar ones. This approach also helps us get out of a rigid, single minded mentality. We open ourselves to other ways of doing and seeing things.

Top Ten questions/comments to Brown & Walter

The Art of Problem Posing by Stephen I. Brown and Marion I. Walter

1. What is the problem asking?
2. Can it be generalized?
3. What are the givens?
4. Are there any exceptions to the rule?
5. What are the parameters and what are the variables?
6. What are some applications?
7. What if the number of parameters was varied?
8. What are the constraints on each parameter/variable?
9. Can the problem be represented graphically?
10. What tools can I use to solve this problem?

In-class timed writing (6min.) My greatest and worst Math teacher ever

Oct. 2, 2019

A) DrMina, my best Math. teacher:
Without doubt, DrMina was my greatest Math teacher. She is the one who inspiredme to pursue a career in science. She simply made Math make sense. Instead of introducing us to concepts, formulas, and equations, she helped us "discover" them as a class ourselves. We had so much fun! We worked on problems together as a team. She read us excerpts from books on Math, told us stories about "i", "0", the history of Math, and Mathematics behind Music, Sports, ... To be able to relate mathematical concepts to everyday life experiences, to see the beauty in the relationships, to be able to make connects, ..., all helped me appreciate Math. and recognize its importance/relevance. I remember feeling comfortable and at ease in class. Discovering the solution of a puzzle or solving a Math. problem was such fun.


B) DrMina, my worst Math. teacher:
I still remember being bored in her classes. I just didn't want to be in that class. I didn't want to take part in group activities she would have us do. I didn't know what the relevance of what we were being taught was to my future. I didn't really care to learn the material.

Aspirations for my future teaching career:
My aspirations are to be able to reach as many kids as possible. To be able to teach effectively and clearly. I would want to be able to convey the message to them that it doesn't matter whether they'll be pursuing higher education or go into careers in math. science, or technology. That learning math. will help them develop their minds and creativity. That it will help them develop problem solving abilities.

Worries/fear for my future teaching career:
That I would not be able to reach as many kids that I would like. That the education system would not allow me or limit my ability to teach the way I would like to. Another worry is that I would not be a creative teacher.

Comments on "Mathematics Education" by Susan Gerofsky

Comments on Battleground Schools: "Teaching Mathematics" by Susan Gerofsky

In This paper, Susan gives us a historical perspective on mathematics education ideologies in North America in the 20th century. She discusses how math. education has oscillated between two polarities: "progressive" and "conservative/traditional". She points out three major reform movements in the 1900s, responsible for shifts between the above two ideologies in math. education:
1- The progressive movement for mathematics through activity and inquiry (circa 1910-1940)
2- The New Math reform movement of the 1960's
3- The NTCM (National Teachers Council of Mathematics) "Math Wars," standards-based math. reforms, from the 1990s to the present.

I find it interesting how questions like "why math.?", "how much math.?", and "how to teach math.?", "how to assess mathematical knowledge and understanding', etc., have been debated and revisited throughout the history. Somehow, I would feel with the age of industrialization, and technological advances, it would go without argument that learning mathematics is essential to function in today's society. Perhaps "how much", "when" and "how to teach" math. are relevant questions to be revisited from time to time, especially in this rapidly changing society.

The assumptions and stereotypes associated with math.and mathematicians is somewhat alarming. Presumptions such as "mathematics is hard, cold, distant and inhuman", or that "it is only necessary and appropriate for a small elite to understand mathematics". As someone who pursued a PhD in particle physics, I admit that the subject material becomes more complex and people pursuing a higher level of education in any field become mainly focused on their area of expertise; however, this does not mean high school math. needs to be difficult to understand and that all mathematicians or physicist are nerds. I think the main reason behind peoples' misconceptions about math. is because math. is not taught well in schools. The reasons behind this as Susan points out are a few: teachers who are math-phobic, teachers who learned math. the conservative way, finding teaching math. in any other way than the conservative way not practical or possible, and unqualified teachers.

The fact that there is not separation between the government/politics and education (as elaborated on in this paper) is not news to me since I know first hand how governments' choice in funding certain projects and not others changes as the political parties in power change. I am not surprised to know how the curriculum in schools is also influenced by the above. I think government's involvement in
setting/shaping the curriculum (in any field in schools) is wrong. Focus has to always be on producing informed, independent thinkers with critical problem solving abilities through the education system and constantly coming up with and implementing methods proven to improve students' understanding and comprehension of concepts based on results from research.

Comments on "Dave Hewitt's" video

In the video we watched, we saw an alternative approach to teaching math. One in which the teachers acts as a facilitator, where he helps students' progress faster and easier by guiding them through activities. Instead of introducing the concepts, he uses activities where students get soaked into and get carried along, starting simple, building to and discovering complex concept through the exercise/activity. Everyone in the class seemed totally focused, interested, and engaged. He was able to introduce the idea of large numbers, arbitrariness of reference points, negative numbers, addition, subtraction, variables, algebraic equations, pattern recognition, ... The process is interactive and participatory. Students seemed so engaged and immersed in the process that they didn't seem to have time to be bored or to misbehave! The class was well-managed and the entire operation well-orchestrated. I wonder how many year of practice it takes for a teacher to get to this skill level.

Assignment #1: Conversations With Math Teachers and Students

http://docs.google.com/fileview?id=0B01xnh0Ekjr6YzM4YWQ5ZTAtZmU3ZS00NzlmLWJiNzgtNTU1MmM1MDc5MWY1&hl=en

Reflections on Robinson's paper on "using Research To Analyze, Inform, And Assess Changes IN Instruction"

http://docs.google.com/fileview?id=0B01xnh0Ekjr6MjFiZGFmNDctYWU3NC00ZDdkLTkxNTctODk2NDU4YTNlNWU2&hl=en

In-class timed writing (6 min.) Write about two of your most memorable mathematics teachers

In-class timed writing (6 min.)
Write about two of your most memorable mathematics teachers


I don't remember the names of any of my math. teachers; Does that mean none were memorable or that I am getting old and forgetful? I do remember two teachers by faces, one a high school teacher (grade 8 or 9) and the other a freshman-year university professor. Its their method of teaching that has stuck in my memory.

The former, sort of combined trigonometry and geometry together. She was very methodical, well organized, and captivating. I still remember to this day how she explained trigonometric functions cos(theta) and sin(theta) in terms of coordinates of a unit circle, with cos(theta) as the horizontal axis and sin(theta) as the vertical axis. In this graphical representation, we could easily find out the values for cosine of simple angles like (0, 45, 90, 180, 270, and 360) degrees could be read right out of the circle. She also showed us how tangent of an angle could be thought of as the line tangent to the perimeter of the unit circle at the end point of the angle.

The latter was also methodical and well organized; however, the way he taught the subject was dry and non-interesting. He did not explain the meaning behind mathematical concepts such as derivation and integration, namely, derivative of a function being the slope of the function evaluated at a given point, and integration as the area under the curve or graphical representation of a function.

Reflection on how this forms my ideas about teaching


I think the above two are examples of relational vs instrumental teaching styles. I definitely hold the belief that 'understanding of concepts' should be the underlying goal while topics are introduced, if not possible at the beginning, then at some later point where connections can be made such that things make more sense and therefore stay with an individual for a long time. To me, this is not
an either or situation, i.e. both methods are valid and there is a place and time for each. The job of a teacher is to flip-flop or go
back and forth, depending on the situation.

Self assessment of my micro teaching assignment: Breath Of Fire in Kundalini Yoga

Self assessment: Microteaching
Topic: Breath of Fire in Kundalini Yoga

I thought these things went well in my lesson:
- Good use of BOOPPPS
- Covering major points of Breath of Fire
- Clear Demonstration

If I were to teach this lesson again, I would work to improve it in these ways:
- Practice giving the lesson few times, I didn't practice at all, just spent time in preparation
- Cover another related type of breathing since I had just under two minutes left. or mention other components of Kundalini yoga in the definition: Krias, Mudras, Mantras, Bhandas?
- Have more comfortable place (yoga mats) for students to sit on
- Learn more about the subject
- Work on staying relaxed

Here are some things I reflected on based on my peer's feedback:
Given that this was the first time I had talked about the subject and taking into account colleagues' feed back, I feel good/positive overall about my presentation. I think I need to work on my confidence since I did not view my presentation as "kindly" as my colleagues did. I felt I could have known about the subject better, communicated better, stayed more relaxed, etc. I think experience in (getting out there) and trying more will definitely help.

Summary of peers' evaluations of my micro teaching assignment on Breath Of Fire in Kundalini Yoga

Colleagues in my group: Amelia, Nathan, Ror, Sam

Checklist (were these elements present?) and Comments:

Learning & possibly teaching objectives clear?
Amelia: Yes, to learn the breath of fire.
Nathan: Yes, great introduction to breath of fire, as well as benefits of it.
Rory: Very clear, I liked how you gave a use for this wake up purpose instead of coffee.
Sam: Clear. Discussed motivation for the exercise.

Bridge/intro?
Amelia: Yes, Mina discussed a bit about yoga, asked if we wanted to feel less tired.
Nathan: Well formed, made applicable.
Rory: Great intro, see above.
Sam: Very good. Motivated by desire to avoid coffee.

Pretest of prior knowledge?
Amelia: Yes, asked if ayone had practiced breath of fire.
Nathan: Yes, history of past yoga involvement/knowledge.
Rory: Well done asking if anyone had done the activity before.
Sam: Asked about prior yoga experience.

Participatory activity?
Amelia: Yes, we all sat in yoga position, practiced breath of fire.
Nathan: Great breathing exercise I can take away and use.
Rory: Great participation. You made yoga in a hallway feel easy.
Sam: Good instructions.

Post-test/check-in on learning?
Amelia: Yes, asked us what Kundalini yoga affects.
Nathan: Yes, a review of the topic and checking.
Rory: Great chekcing on how we felt each round.
Sam: Checked to ensure instructions were understood.

Summary/conclusion?
Amelia: Yes, told us what we might build on next time.
Nathan: Yes, a good closing.
Rory: Very nice activity. Good involvement with everyone.
Sam: Linked to other topics/areas of yoga.

What were the strengths of this lesson?
Amelia: Great demonstration of breath of fire, good participatory activity, informational, good intro!
Nathan: Great participation, great use of time and topic, great opening energy.
Rory: Great activity, great explanations of benefits, great display presentation, very clear and good time use,
enjoyable.
Sam: Clear instructions, good time management, strong intro.

What areas need further work and development?
Amelia: Maybe define the steps more clearly in the learning objectives of breath of fire.
Nathan: Makes us want further lessons from Mina! Need more time!
Rory: Excellent use of BOOPPPS. For a 10 min. intro session, I can only imagine a more comfortable room which was beyond your
control.
Sam: Some minor discrepancies near end with 4 main areas breath of fire aids. Overall, highly enjoyable.

Micro Teaching Lesson Plan on "Breath Of Fire in Kundalini Yoga"

Breath Of Fire in Kundalini Yoga

Bridge: Pose the following question:
Have you ever felt fatigued, exhausted/groggy and wished you could pick up your energy level without the help of a cup of coffee, other kind of stimulant, props, etc?
Well, I am happy to say "there is a way!" :-)

Teaching Objectives:
- Arouse students' curiosity and interest in the concept of mindful and conscious breathing
o Often-times we don't pay attention to how we breathe
o Breathing becomes a task that we do unconsciously - in the background
o Its normally shallow and slow
o We become aware of it only at times of nervousness, high level of anxiety, or when participating in a physically
demanding activity, i.e. running
o There are different breathing techniques that can be used to lower or increase body temperature,
to detoxify the body, to increase energy flow, to relax and slow the mind into a meditative state

- Introduce Kundalini yoga and breath of fire
o Kundalini yoga is a physical and meditative discipline within the tradition of yoga
o It is a subdivision of hatha yoga, more popular style of yoga
o Kundalini yoga helps in moving prana (life force)
o It targets the immune, nervous, glandular, and digestive systems
o Breathing (paranayama) is a major component of Kundalini yoga
o Breath of fire is one breathing techniques in yoga

- Mentiona some of the benefits of breath of fire
o Helps in detoxification and removal of waste by oxygenating blood quickly
o Builds lung capacity and helps purify the respiratory system
o Generates heat and increases level of energy by activating the energy flows in the body

Learning Objectives:
- Students will know the basic definition of Kundalini Yoga and the subsystems it benefits
- Students will find out about the benefits of breath of fire
- Students will be able to demonstrate breath of fire

Pretest:
- Have any of you done yoga? How about Kundalini Yoga?
- Do you know about breath of fire?

Participatory Activity:
- Demonstrate practing breath of fire and aks the students to observe
o Sit up in a comfortable position (easy/cross legged pose)
o Elongate the spine upwards, lengthen the neck and subtly bring the chin back
This will align the spine with the back of the head
o Close your eyes
o Rest your hands in a comfortable position on the knees
o Relax your chest and stomach (diaphragm) muscles
o Now begin to breathe rapidly through the nose with equal emphasis on the inhalation and exhalation
Keep the breath shallow, just at the tip of the nose
Proceed at a comfortable pace and establish a steady rhythm
The stomach will pulse on its own in rhythm to the breath
Continue for 1 minute
o That is it!

- Lead the students to do breath of fire as a group

Post Test: Ask the students:
- What is Kundalini yoga? What subsystems does it work on?
- What are some of the benefits of breath of fire?
- Can you demonstrate breath of fire?

Summary:
- In this class you have:
o been introduced to Kundalini yoga and different breathing techniques in kundalini yoga
o learned about the benefits of the breath of fire technique
o learned how to correctly execute breath of fire and when

-Next class:
o We will learn about Four-part Deep Breathing, another breathing technique in Kundalini yoga: its benefits and how
to execute it

A commentary on ``Relational Understanding and Instrumental Understanding'' by Richard R. Skemp

A commentary on
``Relational Understanding and Instrumental Understanding'' by Richard R. Skemp

In his paper titled ``Relational Understanding and Instrumental Understanding'', Skemp identifies two faux amis in the context of mathematics: 'understanding', where 'understanding' could take on either of the two meanings: 'relational understanding' and 'instrumental understanding', and the word 'mathematics' itself where the 'mathematics' could refer to two different subjects: 'relational mathematics' and 'instrumental mathematics'.

Skemp's statements are based on his observations as to how the subject matter is being taught and learned in schools. His viewpoint is definitely in favor of relational mathematics, where a student is taught not just what to do to get to an answer or how to solve a problem but also why. This is in contrast to instrumental mathematics where the student is taught a set of rules or formulas without reasons. He backs up his viewpoint by listing four advantages for each method of teaching/learning but at the end, he still votes in favor of relational mathematics, saying: ``If the above is anything like a fair presentation of the cases for the two sides, it would appear that while a case might exist for instrumental mathematics short-term and within a limited context, long-term and in the context of a child's whole education
it does not.'' He also goes on to further say: ``So nothing else but relational understanding can ever be adequate for a teacher.'' I fully agree with his arguments and support his conclusions. For individuals to become 'agents of growth', they must develop an understanding of the basic underlying principles. It is through this understanding that they can dig deeper, explore further, make connections between two seemingly different concepts, and become involved in the creative process. Although Skemp uses mathematics as a particular subject area to discuss, I believe the concept is easily carried to other subject areas such as 'physics', 'chemistry', and even 'history'. Its through critical thinking, teaching, learning, and having a content-centered approach to teaching and learning that any of the above subjects can go from being an instrumental subject to a relational one.

Skemp also discusses the possible 'kinds of mis-matches that can occur exist in mathematical teaching/learning'': Instrumental learner and relational teacher, relational learner and instrumental teacher, and instrumental teacher and rational book. I actually could think of another possibility that he has left out, namely, relational teacher and instrumental book. Of all the above mis-matches, I believe the ones with instrumental teachers are the most serious; Skemp call is more ``damaging''. If a teacher's intention, beliefs, knowledge and skills favor relational teaching style, he or she is able to see beyond the limitations of the required text book and reach the few students who really want to understand rationally.

I loved the distinction Skemp makes on the two kinds of simplicity: ``that of naivety; and that which, by penetrating beyond superficial differences, brings simplicity by unifying. It is the second kind which is a good theory has to offer, and this is harder to achieve.'' I would say 'relational teaching' leads to the latter kind of simplicity. i.e. the ability to connect and find the unifying theme. This is what I call ``the joy of understanding'', when the ``Ah-ha!'
' moments happen.