Monthly Archives: December 2011

No fond farewell to 2011

Old year’s day and the ambulance trundles up the driveway in our housing area, a neighbor is taken to the hospital, condition unknown. On Christmas day an outer island community tragically lost a senior pastor.

Shrue Lee Ling
Joy

After Shrue tumbled down the stairs leaving midnight service at the start of 2011, I had a sense the year was not going to go well medically. I told the family that the goal this year was mere survival.

Hanna
Solitude

And the year lived up to its opening shot. One member of the family in hospital for a week including surgery after a gall bladder attack, another one rushed to hospital unconscious. Both would recover, but at the time survival seemed in doubt.

One aunt would pass away – the first of the siblings to pass, an uncle is at present on referral in the Philippines.

A river in Dien
Comaraderie

I would start the year an upset gut – mucus, blood in stools. Three trips to three different doctors had ruled out all forms of dysentery. The third doctor wanted to refer me to the Philippines for a thorough bottom-end physical.

Dien trio
Partnerships

By February I had begun to puzzle out that lactose was problematic. Elimination of lactose helped – no more blood, but rolling into April I was still having problems. A visit by my father-in-law in April saw me feeling better, after he left my problems returned. By May I stumbled across the information that gluten intolerance can cause lactose intolerance, and in an instant I realized that during the one week in April we ate only local, non-gluten foods.

Dien
Confidence

In June I eliminated gluten and my gut became healthy. I could even consume lactose. Seems my childhood allergies and the hyperactive immune system responsible for them are not so gone nor forgotten.

Nihco
Wistful

With 2012 looming, odds of survival through to the new year look better. 2012 will bring its own challenges, triumphs, tragedies, but 2011 started off badly and fulfilled that ominous portend, throwing health challenges at many, and not everyone made it to 2012. As my youngest notes, even numbers are simply better than odd ones. 2012 could go awry, but somehow 2012 holds out the hope of a better year, an even year after all.

Playing Life on the Farm
Playful
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Teacher Corps Assessment

At the end of a week long mathematics and science workshop the 21 participants were asked to respond to the following questions. A report on the workshop exists as two blog articles, Teacher Corps and Teacher Corps II. Responses were obtained from 17 participants.

1. What was the most useful activity for you as a teacher?
2. What was the least useful activity for you as a teacher?
3. What was the most interesting activity?
4. What was the least interesting activity?
5. What was the most surprising experience during this past week?
6. What was the most fun?
7. What would you change if such a workshop was run in the future for teacher corps?

The table below is an excerpt for a larger table of responses. Responses were tallied and common responses were combined. The table includes only those responses which appeared three or more times. Note that respondents were permitted to cite more than one activity per question if they chose to do so.

Response Most useful Least useful Most interesting Least interesting Surprising Fun Sum
Plant names 4 3 2 7 3 19
Field trip 2 3 2 7 14
Floral litmus 1 5 1 2 9
None 7 2 9
All 6 6
Constructions 1 1 2 1 5
El Niño 1 1 2 1 5
Local materials 3 1 4
Speed of sound 1 3 4
Fibonacci ratio 1 1 1 3
Marble math 1 1 1 3

The “plant names” response refers to a number of walks on which plants were identified by the instructor in the local language of the participant. The participants did not all know their own plant names and many found this interesting and surprising.

A field trip to the Pwunso botanic garden to view spice plants, timber trees, and learn about the benefits of local foods from the Island Food Community garnered the most votes for being fun.

A laboratory that used boiled flowers to generate floral litmus solutions the most interesting activity for the participants, followed by a laboratory that determined the speed of sound using sticks from the forest, orally counted seconds, and echoes.

Six of the seventeen respondents felt that all of the activities engaged in were useful to them as teachers and as future teachers. Seven felt that none of the activities could be classified as least useful to them.

Constructing circles, triangles, squares, pentagons, and hexagons with a string and straight edge generated the strongest even split of all activities.

Although not shown in the excerpt above because the number of responses was only two, a side unit on a sound wave done in the computer laboratory, a unit on logic (categorical propositions, the square of opposition, and categorical syllogisms), and a batteries and bulbs activity were the only activities to receive more than one negative response.

The participants were also asked what they would change if the workshop were to ever be run again. The following responses are in descending order of popularity.

Move the start time back one hour from 8:00 to 9:00 (4 respondents)
Ensure lunch is arranged (4 respondents)
Run the workshop at dates that do not fall so close to a major holiday (3)
Cover how to prepare a science worksheet for lower grades (1)
Have more field trips (1)
Shorten the workshop day to four hours (1)
Extend the workshop to three weeks (1)
Spend more time outside (1)

Teacher Corps II

Thursday morning, day four of the workshop, opened with a focus on captivating students’ attention. No attention, no learning. Rather than say this up front, however, the concept was made concrete by putting a teacher, supported by two other teachers, on a RipStik caster board.

With a teacher standing on the board, the difference between dynamic and static stability was explained. Having a teacher being held up on the unstable, stationary board, focused the attention of at least the teacher on the board, if not the class.

With the definition illustrated, the concept was extended to climate change. If the global climate is essentially statically stable, then small perturbations in that system should engender nothing more than small, fairly stable changes in the global climate. If the global climate system is only dynamically stable, then small changes may have unexpected effects including potentially large changes as described in runaway climate change scenarios.

Following this presentation, the instructor used the RipStik to introduce waves. The RipStik leaves behind a distinctive wave on the paper. The wave form provides an opportunity to introduce terminology such as crest, trough, wavelength, and amplitude. The RipStik also makes concrete frequency as being the number of “wiggles” per second.

Dana on a RipStik laying down a waveform
Dana on a RipStik laying down a waveform

Rapid wiggling generates a high frequency (big), short wavelength (small). Slow wiggling generates a low freqency (small),  long wavelength (big). Thus the caster board well demonstrates the inverse relationship between wavelength and frequency that is seen in many systems.

Best of all, for the caster board the wave speed (frequency times wavelength) is exactly the linear board speed.

Images of the tracks with labeled features were illustrated in an article written by the workshop lead in October 2011. The activity is used in conjunction with a unit on waves in physical science.

The board ridden on paper on concrete provides a way to bring wave phenomenon down into earlier grades below the high school level. The boards do cost money, and one has to either ride the board or have a rider, yet there are a fair number of young riders even here on Pohnpei and thus it might be an option for a teacher. Simply have the student ride their board across the paper.

Inside the classroom transverse waves on a length of chain and longitudinal waves in a Slinky spring were demonstrated.

Following the Thursday morning break, the 10:00 session started with geometric math standard 2.31 recognize common shapes. But in a twist on standard 2.8.1, all shapes were constructed using only a length of string as a compass and a meter stick from the forest. The meter sticks had been built on Monday. Constructions based on these limitations are well covered by Zef Damen.

Constructions started with a circle and moved on to equilateral triangles, hexagons, squares, and finally a pentagon. The proof of the Pythagorean theorem was also presented, along with proof of the irrationality of the square root of two. This last fact was problematic for the Pythagoreans and their math system that effectively postulated all one needs to do mathematics are “marbles and pompoms“, along with ratios of marbles and pompoms.

Rustem analyzes acids and bases, Trevor on the right
Rustem analyzes acids and bases, Trevor on the right

In the background above the pentagon/pentagram construction can be glimpsed, on the far right is part of the Pythagorean proof.

The square root of two is not, however, expressable in the Al Mat marbular system, much to the consternation of the Pythagoreans. Thousands of years later Cantor would show that the infinity of irrational numbers is a higher order infinity than that of integers.

By the end of the session the class had moved up from 2.3.1 recognize common shapes and 2.4.1 identify and classify shapes up past 2.8.1 construcitons, 2.8.4 Pythagorean theorem,  1.8.3 square roots, and on into a presentation on the proof of the irrationality of the square root of two.

After the lunch break the class spent a half an hour in the computer laboratory were a fourier sound applet was demonstrated. The applet showed the connection between wavelength and frequency for sound waves, along with a graphical representation of a sound wave.

Then the teacher moved downstairs to engage in a laboratory using floral litmus solutions to detect acids and bases. This was based directly on physical science laboratory thirteen.

Yasko with acid, base, and neutral detection
Yasko with acid, base, and neutral detection

The session served science standards Sci 1.hs.1 and the Sc hs benchmark chemistry bullet item number eighteen, the study of acids, bases and salts.

Benskin
Benskin with test tubes

The laboratory also demonstrated the use of minimal glassware and locally available materials including common household chemicals in a chemistry experiment.

Cheryl tests a floral litmus solution against a known base
Cheryl tests a floral litmus solution against a known base

In the final session of the day which began at 3:00 in the afternoon, the teachers returned to the computer laboratory where science outcomes 3.4.2, 3.5.2, El Niño, La Niña, tropical storm formation, and climatic patterns were presented using presentations put together by Chip Guard of the National Weather Service on Guam. The instructor owes a deep debt of thanks to Chip Guard and the NWS for sharing those presentations.

Friday morning began with a discovery learning session using batteries and bulbs. This particular exercise derives from physical science laboratory 12 and served FSM science learning outcome 2.8.7. A Pohnpei Utility Corporation bill was also worked.

Lorleen measures Deffeny
Lorleen measures Deffeny

Following this session, the class determined their FiboBelly ratio, an exercise selected from a series of exercises centered on numeric patterns including the Fibonacci sequence.

Yasko measures Benskin
Yasko measures Benskin

This exercise is often done in conjunction with Fibonacci factors and Pigonacci. Pigonacci includes the connection between the Fibonacci numbers and Pascal’s triangle.

FiboBelly ratios on the white board
FiboBelly ratios on the white board

The final white board with the teachers’ Fibobelly ratios.

After the morning sessions the teachers prepared workshop evaluations and assessment. Some of the teachers chose to work in the computer laboratory.

A204 Computer laboratory
A204 Computer laboratory

The workshop wrapped up on Friday the 23rd of December with a pizza luncheon and certificates of completion for all participants.

As an addendum to the Thursday afternoon presentation, the following is an account of the damage done by typhoon Lola in November 1957 held in the Pacific Digital Library.

Typhoon Lola Pays A Call

A most unladylike intruder by the name of Lola paid a call upon the Trust Territory in mid-November 1957.

Lola was a typhoon of major proportions. Sweeping along like a bulldozing
broom, she smashed down valuable breadfruit and coconut trees, submerged crops, wrecked homes and generally produced havoc as she rolled on from the Marshalls through Ponape, Truk, Guam, and up to
Rota.

The typhoon which caused more over-all damage than any previously recorded within the territory, brought no loss of life and no major bodily injuries as far as is known, although many times tragedy knocked hard and close. In the face of danger, numerous spontaneous acts of valor came to the fore.

Lola entered Ponape District on November 12, leaving havoc, destruction and debris as she whirled on her way. Not for fifty years had Ponape had a typhoon. It was generally considered to be out of the typhoon path. But reports from atolls and islands throughout the area repeated the story of coconut and breadfruit trees destroyed, and of food shortage imminent after the windfall of nuts on the ground would have been made into copra or consumed for food, and the breadfruit eaten.

Kolonia, the Ponape District center, was in the direct path of the storm, as were the islands immediately around it. Knowing that the typhoon was coming, the people of Kolonia took shelter in the hospital building and warehouse, District Administration office, Intermediate School, agriculture station, and in churches and other buildings of the religious missions. For
some 250 or more storm refugees in these shelters, C-rations (individual canned foods), rice and sugar were issued by the Administration, also small quantities of kerosene to provide fuel for the ranges on which people prepared hot food and beverages.

The damage to buildings and utilities at Kolonia was considerable. Destroyed were the temporary warehouses and carpenter shed on the site of the new Pacific Islands Central School, and ruined was all of the
bagged cement therein, a total loss representing some five thousand dollars.
Ponape’s power and telephone systems were heavily hit by falling trees; roads were eroded, and bridges and culverts damaged, with a loss of approximately thirty-three thousand dollars in government property alone.

In addition to buildings damaged and public works systems affected, four vessels went aground in the bay – all privately owned. These were the LUCKY, the CULVER, the MARU, and the ASCOY. All except the first were expected to be refloated. The LUCKY, which was directly hit and forced high onto the reef, was not thought to be salvable.

Cacao pod production in Ponape was reduced by at least fifty per cent by the typhoon, according to estimates, and copra production here also is diminishing as a result of the high winds which blew immature nuts to the ground, or weakened them so that they began falling off before
ripening.

Teacher Corps Workshop

The week of 19 December marked the start of the Teacher Corps winter workshop with a focus on mathematics and sciences.

The workshop opened at 08:00 with a focus on the FSM national standards and benchmarks in science calling for the use of simple measuring tools (Sci 1.3.3, 1.6.4).  Meter sticks were constructed from locally available wood (Hibiscus tiliaceus, Campnosperma brevipetiolata) in order to measure lengths.

Rulers were used for measures less than one meter.

Mass was measured by suspending the meter sticks from the 50 centimeter mark, suspending a known locally available mass from one arm, and suspending an unknown mass from the other arm. For a known mass the class used bars of hand soap which were labeled as having a mass of 113 grams.

mass balance
Local mass balance

The farther out along the arm the masses can be suspended, the more accurate the mass measurement. As per physical science laboratory one, the volume of the unknown mass was determined by carving the soap into a rectangular slab and then using length times width times height to obtain the volume.

This yielded a density of less than one gram per cubic centimeter for Ivory®  soap, more than one gram per cubic centimeter for the other soaps in the laboratory. This leads to a prediction on whether a given brand of soap will float, a prediction which can be tested.

The Ivory has to be measured as carefully as possible as the density is just under one gram per cubic centimeter.

Density is a physical property, which is a focus of FSM science standards 2.3.1, 2.5,2, and 2.6.3. Note that the first digit refers to the standard, the second digit refers to the grade, and the last digit is the outcome. The standards are:

  1. Number, completeness, and computation
  2. Geometry, measurement, and transformation
  3. Patterns and algebra
  4. Statistics and probability

At 10:00 the focus shifted from science to the precursors of mathematics. The teachers were introduced to the basics of set theory in order to prepare them for Al’Mat and Al’Jabr the next day.

Set theory and the presentations that would follow provide examples math standard 1.3.4, representing whole numbers using physical models and diagrams, and the extension of those models into arithmetic and algebra.

After a lunch break the teachers were tasked with counting the pillars and posts on campus – at least all of the pillars and posts for the buildings and walkways connected by covered walkways.

Covered walkways

The four groups of teachers returned with the numbers 250, 267, 286, and 364. The object was to demonstrate the inherent “fuzziness” of even a simple measurement. The teachers all averred that there is a correct number of pillars, but they were unable to agree on the actual value.

The exercise also brought to the fore the need to specify definitions prior to gathering data.

The lack of a single value led naturally to a discussion of the role of range, mode, median, and mean as ways of characterizing a data set. While the college posts and pillars are a handy, countable item, the teachers will have to select other locally available numerous items to count if they choose to replicate this activity. The activity was based loosely on an activity I did with the Upward Bound students in the summer of 2007.

The statistics unit address math standards 4.4.1 collect data, 4.5.2 organize data using tables, charts, 4.6.1 range, mode, median mean, and 4.8.1 choosing the best measure of middle.

The planned end of day hike was rained-out and evening was settling in, hence the workshop dismissed for the day.

Tuesday morning the teachers measured the speed of sound using nothing more than their homemade meter sticks, counting the seconds orally, and two blocks of wood. The exercise was a modification of physical science laboratory nine.

With meter sticks: Tracy, Aileen, Gracelyn
With sticks: Tracy, Aileen, Gracelyn

Above the teachers are laying their meter sticks end-to-end to measure the distance for the echo flight.

Deffeny lays out the three meter stick
Deffeny lays out the three meter stick

Bear in mind that the meter sticks were constructed by first using a commercial meter stick to determine a distance on the teacher’s own body that is one meter. This was then their standard for building their own meter sticks. If nothing else, one has one’s body as a starting place to make new meter sticks in the future.

Gracelyn adds sticks as Randy, Aileen, and Tracy look on
Gracelyn adds sticks as Randy, Aileen, and Tracy look on

The total distance from the building to the road was 80 meters. The sound echoes produced traveled double that distance, 160 meters. While one teacher counted “one-one thousand, two-one thousand, three-one thousand,…” up to ten seconds, a second teacher clapped two boards in synch with the echoes. A third teacher counted the claps.

The view along the line of sticks
The view along the line of sticks

On one of the trials the teachers counted 23 claps of the wooden boards – 23 echo flights – in ten seconds. That means a flight time of 0.43 seconds per echo. Dividing 160 meters by the flight time yields an estimated speed of sound of 368 meters per second.

At the air temperature on Pohnpei, sound travels at 350 meters per second, give or take two meters per second. This means that using only sticks from the forest, oral time counts, and two boards, the teachers obtained a result accurate to within 5%.

The speed of sound unit served science 1.3.3, 1.6.4, 2.6.3, and a blend of 2.8.3 describe sound coupled with 2.8.4 kinds of motion.

With the weather looking good, the teachers slipped in a hike down the road to look at ferns including Davallia pectinata (syn Humata banksii). The fern is presently identified as being either limwediliniak or kelmahu. The fern is not one the students had ever noticed before – at a glance one might think the fern is Dicranopteris linearis (mwedil en mal), but the “pectoral” fins on the D. pectinata are distinctive. In addition, D. linearis is terrestrial, D. pectinata is epiphytic. Other plants were also covered on this walk, with support material available in the ethnobotany text. This unit serves science outcome 4.3.4, identifying common plants around the school.

Just after 10:00 on Tuesday the class moved on into Al Mat, the mathematics of marbles. This unit sought to make concrete concepts such as identity, closure, commutivity, and distribution. This section served math standards 1.3.3 understanding arithmetic operations, 1.3.4 and 1.4.3 represent whole numbers using physical models and diagrams, as well as 3.6.2 represent patterns pictorially. The unit included a brief introduction to Al Jabr, although this topic would be further developed on Wednesday.

The teachers broke for lunch. Lunch was followed by a fifty minute high-speed introduction to botanic diversitry, charging from mosses through seedless vascular plants, up through gymnosperms and on into angiosperms. The teachers were also introduced to the college herbarium and the information contained therein. This and the following field trip served science outcome 4.5.2, distinguishing plants with and without seeds.

This was followed by a field trip that hit three broad areas in a single two hour session. The teachers toured the Pwunso botanic garden and were shown economically important plants such as clove trees, cinnamon trees, coffee, black pepper, nutmeg, allspice, mahogany, kauri pine, cook island pine (timber tree), and teak.

Hattie smells the cinnamon tree
Hattie smells the cinnamon tree

Note that the cinnamon trees might easily be mistaken for madeu, the trees are, however, Cinnamomum verum, not Cinnamomum carolinense.

Deffeny
Deffeny at the cinnamon tree

The teachers were also specifically introduced to the gymnosperms present at Pwunso including Cook island pines, a large cycad, and the kauri pines.

Emihner Johnson
Emihner Johnson

The third leg of the field trip was a presentation by Emihner Johnson of the Island Food Community of Pohnpei on the CHEEF (Culture, health, environmental, economic, food security) benefits of local food.

Hattie Peter, Lillyrose Nesheim, Jessica Herry
Hattie, Lillyrose, Jessica

Wednesday morning the teachers engaged in a laboratory designed to serve mathematics outcomes 3.7.3 locate points on the coordinate plane and 3.8.2 graph linear functions in two variables using a table of coordiante pairs. This exercise was based on physical science laboratory two.

While the physical science laboratory utilizes a ramp to produce reproducible speeds, the activity Wednesday morning was done on the front porch of the A building using a bowled four square ball. Like the physical science laboratory, the data was plotted loosely on poster pad, but with a twist.

Deffeny with the time-person coordinate plot
Deffeny with the time-person coordinate plot

In the first phase time was counted orally (x axis variable), but distance was denoted using only the names of the teachers. The teachers stood at the “second” mark as per physical science laboratory two, but the y-axis was initially just their names spaced roughly as they had been spaced on the front porch.

Time versus Dana, Deffeny, Cheryl, and Trevor
Time versus Dana, Deffeny, Cheryl, and Trevor

The teachers were then introduced to “marbles in equals marbles out” and “speed in equals speed out” based on physical science laboratory four. The teachers at first proposed that “force” or “energy” accounted for the behavior of the marbles, but when pressed none could use their term in an explanatory fashion.

Too often “magic words” are used to “explain” phenomenon without understanding what the magic words mean. Magic words also often paper over the deeper mystery of a system, such as how the marbles keep count. The teachers, fascinated by the marbles, spent rest of the first morning session chasing ideas about why the marbles were behaving as they did.

At 10:00 the grounds crew began cutting grass with weed wackers (string trimmers) on the south side of the building while the electrical crew began drilling concrete on the north side of the building. This provided a perfect excuse to go peripatetic, and the teachers wandered off to the front of the library for an Aristotelian peripatetic school presentation on Al Jabr, Categorical Propositions, the Square of Opposition, and Syllogisms.

Materials for Al Jabr were spread on a towel on the sidewalk. Logic was presented purely orally, in keeping with the theme. As a help to the Peripatetic pupils, handouts were also provided.

The session was too brief to properly cover such a broad range of topics. Also covered was being adaptive in one’s teaching. When the classroom became unusable, the class moved. Education can happen without a classroom, without supplies, without blackboards and chalk.

The Federated States of Micronesia and the education systems therein face an uncertain financial future at best, severe budget cuts year-on-year going forward to 2023 and beyond. Should the nation suddenly stumble into unexpected wealth, then teachers will likely have dream classrooms. At present, however, the financial outlook for the schools is bleak. Teachers will have to learn to make do with whatever they can scrounge up, cobble together, and create. One of the aims of the workshop is to begin to provide exactly those tools to the next generation of teachers.

As in the days of Aristotle walking and lecturing in the agora, the listeners became distracted by the aroma of lunch and the class broke for their midday meal. Thanks are due to the teacher corps coordinator for arranging lunch, this provides an opportunity for the workshop to socialize together over a meal, including the instructor who also chose to eat in the cafeteria.

After lunch on Wednesday we looked at CD spectra boxes, colors, spectral lines, and RGB generated images.

Spectrum discharge tube
Spectrum discharge tube

This unit started off serving science outcome 1.3.1 make observations, with the observation initially being the colors of the solar spectrum as seen in a CD spectroscope. In the CD spectroscope image below the lines of helium can be seen.

Helium lines
Helium lines

While helium discharge tubes are obviously not going to be available, the teachers were then instructed to point their spectroscopes at the fluorescent lights. The fluorescent lights also produced a discrete set of spectral lines. These lines were connected back to electron orbitals – a physical property of atoms – and to the development of quantum mechanics.

In the last session on Wednesday the class went on anther hike to look at more local plants including lycopodium and a variety of healing and food plants.  This session continues the work on serving science outcome 4.3.4, identifying common plants around the school.

Walking in from the botanic garden
Walking in from the botanic garden

Hawaiian kava

Pohnpei has two varieties of sakau,  rahmedel (rahmoadoal), and rahmwanger. Rahmedel has smooth stems, a light green stem color, and long internodes. Other names for rahmedel include kohre and kalaidong. Rahmwanger has darker stems, black spots on the stem (not smooth), and short internodes. Other names for rahmwanger include kohkore and nahniepw (nahnioapw).

Kosrae has a third variety of sakau with a different mix of kavalactones.

Kava variety from Hawaii

Last night a local market served a variety said to have originated in Hawaii. The market had already determined that the variety was bitter beyond the capacity of the keleu (koaloau) to offset. As a result the market chose to blend the Hawaiian kava with a local variety.

Kava variety

The internodes are very short and the stems do not develop black mottling spots until they are more mature. The spots are not as prominent as those on rahmwanger. The plant also appears to grow taller than the local varieties on Pohnpei.

Kava stem

The blend has a peppery undertone and strong almost tropical woodsy aftertones. The flavor is quite different from that of the two varieties present on Pohnpei.

Kava stem
Some internodes are longer

Because the mix was a blend, determining which effects were due to the local plant and which due to the Hawaiian plant could not clearly be determined. The Hawaiian kava component did seem to lend strength to the sakau which remained strong into the evening.

The natural emetic effect of dihydrokavain and dihydromethysticin complicates drinking strong kava, my own sense was that the Hawaiian plant had higher concentrations of DHK and DHM than, for example, the high kavain variety of Kosrae.

The rapid onset of sleepiness followed by less severe oan sakau suggests more DHK than DHM, but again, the brew was a mixed batch. The mixed brew certainly renders one sakaula and diplopic fairly quickly and I was home and sleeping like an infant before eight in the evening.

Site Swap Laboratory

SC 130 Physical science laboratory 15 is an end-of-term exercise that attempts to expand the students’ mathematical horizons beyond the algebraic focus that permeates the earlier laboratories. Site swap mathematics provides a different mathematical model that also yield testable predictions.

The course started on a Freeman Dyson quote about the mathematical nature of the physical world. Nature obeys mathematical equations, and, in turn, those equations predict what will occur or can be done. Site swap notation provides another approach to this concept. Swapping sites provides another pattern that can be tested by juggling the new pattern. There are true site swap “equations” and false site swap equations.

334233 is a true statement. 25 is a false statement.

The laboratory also ends the term on an activity that is simply fun. Leave them laughing if possible.

After explaining site swap notation, the students get a chance to learn to juggle. Merna picks up the skill quickly.

 

 

 

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