Saturday, February 27, 2010

Snapshots of Navigation in Guam and Chuuk

As we've progressed in our coursework, some of you have shared some interest in mathematics dealing with canoebuilding and navigation. The mathematics involved include the required amount of time, effort, materials, proportions, and measurements needed in the making of a canoe and its various parts. Some things to consider when making a canoe are the hull, sail, boom, stanchion, crosspiece, outrigger platform, longitudinal stinger, diagonal stinger, removable board, weather-side board, and thwart.

Guam's history of navigation dates back to the flying proa - sailing vessels - which required intuitive construction. The canoes were the ancient Chamorros' primary transporation between the Mariana islands for trading, war, recreation, and match-making. Sailing techniques were used by them to navigate to and from islands with speeds of up to twenty miles per hour. There are different canoes, such as the ladjak, which referred to canoes that could be fitted with sails, with the sakman as the largest, followed by lelek, duding and duduli. There is also the simplest of canoes called the galaide' - a dugout paddling canoe. Our ancient Chamorros even built canoe houses to protect the canoes. Unfortunately, by the 1900's the tradition of the ladjak was lost and only the galaide' remained, although that almost became extinct.

Today, a revival project has generated classes, projects, and groups who remain committed to bring back the proud seafaring traditions in Guam and the Marianas. As part of the revival project, a few of our dedicated Chamorros sailed to Polowat Atoll, Chuuk, where traditional seafaring knowledge is still alive. Polowatan navigators have also visited Guam to teach our younger generations how to build canoes.

We are grateful that our fellow Pacific islanders in Polowatt Atoll, Chuuk, have helped us to revive our ancient building and navigation of canoes. As such, it is only fitting that I share some navigational snapshots from both Guam and Chuuk. The Guam snapshots were retrieved by Guampedia. The Chuuk snapshots were contributed by Richardson "Rick" Chiwi, COM-FSM Instructor, who was a dedicated student of mine for ED638: Teaching with the Internet.

Guam Navigation Snapshots...

Chuuk Navigation Snapshots...

ED 638 is the vast ocean to tame and conquer; with patience, persistence, and perseverance one may overcome all odds and be rewarded with the many resources it offers. Wind, current, stars, and the waves are guidance to navigate with. Blogger, ClassCentral, HotChalk are guidance to ED 638. So much to thank about!
Posted by Richardson Chiwi at 9:26 PM 1 comments

WASC visiting member, Mr Yamaguchi of Hawaii visited our campus and was welcomed to set foot on our canoe.
Posted by Richardson Chiwi at 8:32 PM 1 comments


Artero, A.D, ' Canoe Building', referenced February 27, 2010, © 2009 Guampedia™, URL:

Chiwi, R. (2009). ED638 - Summer 2009 Blog. Retrieved February 28, 2010 from

Sunday, February 14, 2010

Mathematics used in singing in the Chamorro culture

Kuma kanta (singing) was a way of life for ancient Chamorros, and it is a practice that is still actively practiced. Whether it be to sing magof pot triste (happy or sad) songs, the very nature of singing has much meaning. For instance, Chamorro fishermen sang songs to bring them luck, and Chamorro lovers would sing songs to warm the heart of their loved ones. We also sing songs in our churches, in storytelling, fiestas, and dancing. The songs reflect how we feel and sometimes point out a moral that is meaningful in our culture.

Mathematics concepts: Rhythm, beat, high & low-pitch notes, timing of notes, amount of words used, length of the song, etc.

Here's a singing mathematics lesson plan that I came across, which will certainly engage the students:

Singing Math
Music Lesson

Objective: Students will recall the twelve times tables up to 120 10 out of 10 numbers.
Math Objective:
Students will acquire number sense and perform operations with whole numbers, simple fractions, and decimals.

Music Target:
Perform basic beat activities using body percussion.

Materials: white board, dry erase marker, tape (not necessary)

"Who knows the song Pop goes the Wesel? Today we are going to learn some of the 12 times tables singing to the tune of Pop goes the Wesel.

I Do: Teacher will write the numbers on the board, clustering together the numbers that will be said together. For example: 12,24,36...48 & 60...72,84....96,108,1 2 0. "Students, clap along with the beat while I sing the numbers out loud." Teacher will then sing out loud while clapping along with the beat.

We Do: Students will join in with the teacher as they begin to get the beat. Teacher will then remove her assistance and students will practice the song with one another. "Remember to clap with the beat, this will help you sing the numbers at the right speed and on the right beat." Teacher will walk around as students practice the song and clap with one another. Teacher will slowly erase numbers off the board, allowing students to use memory instead of reading the numbers. By the end of We Do all numbers should be erased.

You Do: Students will practice independently, using clapping if needed. Students will then sing the song to the teacher individually. Students can clap the beat in order to help them sing their 12 times tables but it is not necessary for their grade. Teacher will observe students as they pass them off.

Assessment: Students will sing the twelve times tables to the teacher.

Weber State University. (2009). Retrieved March 4, 2010 from

Mathematics Used in Chamorro Celebrations

On the island of Guam we have many fiestas for various celebrations, including the feast for patron saints, baptisms, weddings, anniversaries, etc.

Mathematics concepts for a fiesta include the following: Amount of food needed for the respective fiesta; consider the space available for guests (i.e. food set-up, guest area, dancing ground); count the number of chairs needed for guests; time needed and amount of ingredients necessary to prepare the various type of foods, including kelaguen, pan (bread), steaming red rice, fish, beef, pork, escabeche (fish salad), titiyas (tortillas), barbecue chicken, fina’denne’ (hot sauce), etc.

In particular, the following are some Chamorro recipes used to prepare delicious dishes served at fiestas. These recipes clearly involve mathematics. Since Chamorro month is coming up, it would be great to have a cooking demonstration for the students and have them be actively involved in the cooking process under your close supervision.


(Beef with eggplant and coconut milk)

1 ¼ pounds flank steak, chopped, or
1 ¼ pounds ground beef
6 to 8 eggplants, parboil, sliced lengthwise
2 cups coconut milk (1 cup very thick, 1 cup very thin)
2 tablespoons lemon juice
1 small onion, minced
1 tablespoon garlic, minced
¼ teaspoon salad oil
¾ teaspoon salt

1. Saute garlic and minced onions in salad oil.
2. Add flank steak or ground beef and simmer for 10 minutes in covered saucepan.
3. Add lemon juice and continue cooking for 20 minutes.
4. Add salt and pepper.
5. Continue cooking for 20 more minutes.
6. Add thin coconut milk and continue cooking for 10 more minutes.
7. Add eggplant and cook for 3 minutes.
8. Add thick coconut milk and simmer for 3 minutes.
9. Remove from heat and serve while warm.

*Note: Makes 4 to 6 servings.


2 ½ to 3 lbs. whole chicken fryer
1 tablespoon salt
½ cup lemon juice
¾ cup chopped onions
1 cup grated coconut
6 small hot peppers

1. Cut chicken in half, sprinkle with salt.
2. Place in a broiling pan for 10 minutes.
3. Turn chicken over and broil for 5 minutes more.
4. Remove from broiling pan and cool.
5. Debone chicken and chop into fine pieces as desired, making about 5 cups.
6. Place chopped chicken into medium size bowl and add all other ingredients, except coconut, and mix well.
7. Add coconut and mix just before serving.

*Note: Makes 6 to 8 servings


5 – 6 bananas
1 cup melted butter
1 ½ cup sugar
2 teaspoon baking soda
1 ½ cup carnation milk
1 cup mixed nuts (preferably walnuts)
1 teaspoon salt
2 eggs
2 cups flour
2 teaspoon vanilla

Instructions: Mix ingredients, and bake at 350 degrees for 30 to 45 minutes.


¾ of butter
2 cups flour
1 cup sugar
2 ½ teaspoon baking powder
2 eggs
2 ½ teaspoon vanilla
1 cup milk
¼ teaspoon salt
½ cup of raisins

Instructions: Place ingredients in a greased pan, and bake at 375 degrees for 30 to 35 minutes.


3 lbs. chicken
2 tablespoon vinegar
1 teaspoon black pepper
½ cup soy sauce
4 large chopped garlic cloves
½ onion, sliced

1. Cut chicken into serving pieces and par boil to remove excess fat.
2. Rinse and drain chicken pieces.
3. Combine remaining ingredients and pour over medium heat until chicken is tender and sauce thickens.
4. Cook for about 30 to 45 minutes.

(Fish with vegetables)

3 ½ to 4 lbs. fresh fish
1 lb. or 25 stems and tips of cadagan
2 cups or 70 tips kankong
6 stems Chinese cabbage
1 large tomato, sliced
1 large onion, sliced
4 cloves garlic, minced
½ cup white vinegar
2 teaspoon sugar
2 tablespoon salt
2 (about 2”) yellow ginger

Pre-preparation for cadagan:
1. Clean cadagan stems and tips by removing fiberous material from stem.
2. Rinse.
3. Bring about 1 qt. water to a boil and add cadagan leaves and tips and continue boiling for 3 to 8 minutes.
4. Remove leaves and tips from water and place in colander.

Pre-preparation for kangkong:
1. Clean and rinse kangkong leaves.
2. Bring 2 cups of water to a boil.
3. Add kangkong tips and continue boiling for 5 minutes.
4. Drain.

Pre-preparation for cabbage:
1. Bring 2 cups of water to a boil.
2. Add 1 ½ to 2 ½ inches cut cabbage. Cabbage should be sliced crosswise.
3. Add cabbage to boiling water and continue cooking for 5 more minutes.

Pre-preparation for fish:
1. Clean fish.
2. Sprinkle salt and pan fry until cooked and crispy.
3. Take ¼ cup oil from pan in which fish had been friend and pour into a medium size saucepan.
4. Add minced garlic and onion.
5. Saute until nicely browned.
6. Add tomato and cook until tender.
7. Add grated ginger and cook for about 2 minutes.
8. Add vinegar, 1 cup water, salt, and sugar.
9. Bring to a boil and continue cooking for 5 minutes.

Final Instructions:
1. In a pyrex casserole dish, place cooked vegetables in alternating layer with fish.
2. Pour cool liquid mixture into vegetable dish.

*Note: Makes 6 to 8 servings
Do not use stainless steel, aluminum or plastic dish.


Egg Batter:
1 egg
½ cup evaporated milk
½ cup plus 1 tablespoon flour
1 teaspoon white pepper
1 teaspoon salt
½ teaspoon garlic powder.

Additional ingredients:
1 cup chopped celery
½ cup chopped green pepper
½ cup chopped onion
1 lb.shrimp (cleaned, deveined, chopped), approximately 1 cup
1 10 oz. package mixed vegetables (chop green beans)
7 oz. can corn
Salad oil for deep fat frying

1. Combine celery, green pepper, onion, shrimp, mix frozen vegetables can corn, and add egg batter.
2. Heat oil in frying pan to medium heat.
3. Drop mixture by tablespoon into the deep fat 1 ½ to 2 in. 350 degrees, until golden brown about 5 minutes.

*Note: Makes 4 dozen 1 ½ in. patties


Feel free to use any of the aforementioned recipes. This comes at an opportune time to share this with the upcoming Chamorro Month activities in the Guam Department of Education. Biba CHamoru!


Naputi, Joaquin. Personal INTERVIEW. 23 February 2010.
Naputi, Julia. Personal INTERVIEW. 23 February 2010.
Women’s Group. Favorite Guam Recipes.

Measurements used in cooking Chamorro food

The focus of this is kelaguen, as it is one of Guam’s most delicious fiesta food.

Mathematics concepts: Amount & size of ingredients used, including lemon juice, coconut meat, salt, donné (hot pepper), chicken (or beef, deer), and time of meal preparation.

Mathematics used in building ancient Chamorro homes

The focus of this is the latte stone, as it was used as pillars for the foundation of homes.

Mathematics concepts: weight, size and shape of latte stones; number of latte stones used for the foundation of the homes

Chamorro Calendar

The ancient Chamorros measured the year from harvest to harvest. According to Cunningham (1992), Sakkan (year) means harvest. He said that the year was divided into thirteen moons, and that each moon seems to have been associated with something that was happening at that time of year. For instance, he said that Umatalaf was the time to catch red snappers (gatafe). This further reminds me of an interview I had with my father regarding how the full moon is a sign for our Chamorro farmers to pick land crabs, as there would be an abundance that would come out during that time of the day. This was part of a lunar calendar that our people used.

The Ancient Chamorro Calendar included the following:
January: Tumaiguine
February: Maimo
March: Umatalaf – to go catch gataffe (red snapper)
April: Lumuhu – to go back, to return to the attack
May: Makmamao
June: Mananaf or Fananaf
July: Semo
August: Tenhos
September: Lumamlam or Lamlam
October: Fanggualo’ or Fa’gualo – planting time
November: Sumongsong
December: Umayanggan – season of slight but frequent showers
Umagahaf or Omagahaf – to get crayfish

Cunningham, L.J. (1992). Ancient Chamorro Society. Honolulu, HI: Bess Press.

Mathematics used in Chamorro agriculture

Many Chamorro farmers know that when it’s full moon, the land crabs come out, so it’s the best time to hunt for them. During this time, the female crabs come out in plentiful amounts, and therefore your catch is bound to be a good one.

My father recalls the days when he would hunt for crabs during the time of the full moon and how he and his family would reap the rewards of their bountiful crab hunting. He mentioned of how he would get home with much enthusiasm, and how family members would grind the coconut, boil the coconut milk with the crab, cook hot rice, and then they would all sit down to eat together and count how many crabs they caught.

So what are the mathematics concepts involved in crab hunting? For one thing, the full moon is part of the lunar calendar. Secondly, on the day of the full moon, we are able to see that various living things feel and act differently, such as the case of the crabs that come out during this time. In some cases, the reproductive cycle is in sync with the moon's cycle. As the moon goes through its different phases we can see the different shapes that it takes, including the new moon, crescent moon, first quarter, waxing gibbous, full moon, waning gibbous, last quarter, another crescent, and then another new moon. All of these are concepts of mathematics with the different stages that occur, and that's not including the fun fact of counting how many crabs were actually caught!

Aside from land-crab hunting, there is the clam digging during low tide on our Tapon Dia (Clam Day). Growing up I would go to the beach every weekend with my family to dig for clams. My siblings and I would use our fingers to dig deep into the sand and pick as many clams as we could all day long. We’d often count the amount of clams we each gathered. It was like a contest for us, but that didn’t matter once we reaped the rewards of our efforts. The reward of eating what we had caught was a special time for all of us.

My mother or father would grind the coconut, so that we could boil the clams with coconut milk and eat it with hot rice. I can honestly say that after a hard day’s work, the harvest of clams, and then eating them, was well worth the effort.

Going back to mathematics, what exactly are the mathematics concepts involved in clam harvesting? For one thing, the digging occurs during certain times, most especially when it's low tide. Secondly, the time of the year or particular months/seasons of the year must also be considered. A mathematical formula is also used to determine the number of clams available, and we must be sure to harvest in a way that ensures we do not deplete the standing stock of clams. All of these elements deal with mathematics, and that's not including the fun fact of us counting how many clams we actually harvested!

Naputi, Joaquin. Personal INTERVIEW. 6 February 2010.

Saturday, February 13, 2010

Mathematics Used In Chamorro Medicine

The focus of this is the Suruhana, as island residents sometimes seek the medicinal/spiritual powers of her when they fall ill. In my family the spiritual stories remain alive of my great grandmother, Antonia Chargualaf Nangauta, who served as the Suruhana (female herb doctor) of Malesso (Merizo). My father shared of how she would cure the sick by giving them herbs to eat or drink, or maybe even massage them. Yet he said that when she healed someone, the sickness imposed by the spirit would transfer to her, so she, herself would have to drink herbs to cure herself. He said that she often instructed those inflicted with sickness to visit the site where their illness began and ask for forgiveness for disturbing the spirits.

Both of my parents reminded me that if I should ever enter the jungle to be careful not to disturb the taotaomo'na (ghost; people of before). They mentioned that I must ask for permission from the Guela yan Guelo (grandmother and grandfather) before I begin my travel, so that I may not get sick, pinched, get red marks, or even swell. Although the Suruhana is revered, my mother said that there aren't as many that can be found these days, for they too, grow of old age, just like my great grandmother, who has since passed away.

Mathematics concepts that my great grandmother had to consider include the following: Amount of herbs used, type and texture of herbs used, amount of water to boil herbs, time of preparation of herbal medicine, number and pressure of massages used to heal, and length of massage therapy.

Naputi, Joaquin. Personal INTERVIEW. 7 February 2010
Naputi, Julia. Personal INTERVIEW. 7 February 2010

Tuesday, February 9, 2010

Chamorro Counting System

While researching on the Chamorro counting system, I came across the ancient Chamorro measuring system and measurement of length that I've only heard a few times in my life and only bits and pieces of them. Cunningham (1992) did a good job in explaining these measurements, as shown below.

Measuring System –

Measurement of dry volume

chupa = 1 cup

ganta = 8 cups = one-half gallon

Measurement of length

dedo = length of the second joint of the index finger

hemi = the length from the tip of the index finger to the spread thumb

kodu = the length from the elbow to the end of a clenched fist

kuatta = the length from the end of the thumb to the end of the little finger when the hand is spread

bara = arm length from the shoulder to the end of the fingers

brasa = length from finger tip to finger tip, with both arms outstretched


Cunningham, L.J. (1992). Ancient Chamorro Society. Honolulu, HI: Bess Press.

Monday, February 8, 2010

Coconut Weaving in the Chamorro Culture

Weaving is said to be a lasting Chamorro artistry. There is mathematics used in the weaving of coconut leaves, for cultural master weavers must count the number of leaves they use, count in patterns, consider the size of the leaves, and dedicate time to the craft being weaved.

My mother is a weaver and has shared with me that leaves have to be wide, so that the craft you make will look more attractive. She said that if the leaves are too skinny it will not cover everything and you will end up having holes in your product.

Among the crafts my mother is able to make within minutes include the following: hat, bird, roses, fan, and headband. She said that it takes about half an hour to weave a hat and bird, 5 minutes to weave 3 roses, 15-20 minutes to weave a fan, and 5 minutes to weave 3 headbands.

She said that the weaving of hats follow a certain pattern, and your counting depends on the size of the hat you wish to make. For instance, the hat for kids would be a smaller size, and it would require about 16 leaves and less counting. On the other hand, the hats for adults would require about 20 leaves and more counting.

My mother said that the headband requires 3 leaves to make, and you would simply adjust it to fit either the child or adult. She said that the bird and rose each require one leaf to make.

She said that counting is especially important when making hats and fans. “You have no choice but to count,” she said. “The fan could be any size, either big or small, but no matter the size, you still have to count on each side for the fan.”

When asked why she enjoys weaving, my mother said that it “feels good and it’s therapeutic. I’m happy knowing that I’m sharing my gift with others.” I am touched by her words, and indeed, I agree that there’s no better reward than knowing that you have made a difference in some way or another. Moreover, I am a teacher because I also want to make a difference, one student at a time!

Naputi, Julia. Personal INTERVIEW. 6 February 2010.

Saturday, February 6, 2010

Mathematics used in farming in the Chamorro culture

For Assignment #3, we were tasked to further specify the mathematics concepts used in our topics. Coming from a family of farmers, I've decided to expand on the mathematics used in farming in the Chamorro culture. If you don't know by now, my father is a farmer, and his father was a farmer, and his father's father was a farmer, and so on. My father shared with me that when you farm, you must consider the distance feet between hills (i.e. plants planted two per hill), the number of plants per hill, the distance feet between rows, the number of plants/100 ft. row, the number of plants per acre, 100 ft. row yield (lb.), and eventually, you'll have to figure out the estimated cost per 100 ft. row, per acre yields (lbs.), estimated cost per acre, and the breakeven price. The University of Guam Cooperative Extension Service has also researched and provided outreach to farmers on the elements I've mentioned.

For instance, the following is a breakdown provided by the UOG Cooperative Extension Service for eating and cooking bananas.

Recommended distance ft. between hills - 10
Recommended no. plants per hill - 1
Recommended distance ft. between rows - 12
No. plants/100 ft. row - 10
No. plt./acre - 363
100 ft. row yield (lb.) - 500
Estimated cost per 100 ft. row - $229.75
Per acre yield (lbs) - 17,500
Estimated cost per acre - $8,340
Breakeven price - 48 cents

Just the figures above involve a lot of mathematics, and the more bananas you plant, the more calculating you'll have to engage in! My father stated that although he loves planting bananas, which take about 14 months to grow, once you cut them down after they bear, they do not bear again, compared to other plants. On the other hand, he said that cucumbers tend to bear more, because once they're ready for picking, you would harvest them every other day. Thus, it takes cucumbers 45 days to grow with the harvesting to occur every other day for 2 weeks.

My father mentioned that if you want to plant long-term fruit trees that always bear fruit, you would have to plant any of the following: mango, coconut, star apple, and sour sap. He said that these plants would ensure that you always have something to harvest once they start bearing. However, he also mentioned that these trees take about 7 years or more to bear. Despite this, the beauty is that once they start bearing, they bear non-stop.

My father always encourages fellow farmers to use organic dead leaves and other compost mater to fertilize the plants, so that the plants can "produce better, grow better, and grow healthier."

When asked about his final thoughts on Mathematics in agriculture, he said that "It (mathematics) is important in agriculture, because it tells you what crops can bring you better economic benefits and can sustain your family for the long run." He also mentioned about the hard work, care, and appreciation that goes into the love of farming.

I am grateful that I was able to interview my father. He provided much insight, and I look forward to conducting more research on this, and how it can impact student learning in mathematics for all grade levels.

Naputi, Joaquin. Personal INTERVIEW. 5 February 2010.
UOG Cooperative Extension Service

Tuesday, February 2, 2010

Number Similarities in Pacific Island Languages

Hafa Adai! It is interesting to note that there are number similarities in Pacific Island languages. In fact, this was noted by H. Costenoble's "The Family Tree of Chamorro," and Cunningham (1992) also reported this in his book titled, "Ancient Chamorro Society."

Here's a snapshot of these number similarities:

Note: Click on the image above to get a clearer view.

I look forward to engaging in more research that will provide additional details on the similarities.

Costenoble, H. (1974). The Family Tree of Chamorro. Guam Recorder, 4(2), 25.
Cunningham, L.J. (1992). Ancient Chamorro Society. Honolulu, HI: Bess Press.