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.