One of the most visible mathematical aspects of rice farming in the lingko lodok system of the Manggarai community is reflected in the distinctive spider-web shaped configuration of the agricultural fields. The Manggarai people refer to this traditional spatial form as lingko lodok. The lodok rice-field structure of the Wae Nakeng community represents a concrete expression of ethnomathematical knowledge consistent with Bishop’s (1988) universal mathematical activities embedded in cultural practices. The geometric structure reflects activities such as designing, locating, measuring, and explaining. The designing activity is visible in the creation of a radial geometric pattern extending from the central point to the surrounding circle, forming a spider web-like configuration. The locating activity emerges through the spatial reasoning involved in identifying the central point and drawing radial segments in specific directions, demonstrating an implicit understanding of geometry and orientation. The activity of measuring found in determining the radius of the inner circle and the divisions along its circumference using the traditional moso unit. Furthermore, the explaining activity is evident in the transmission of the philosophical meaning of the lodok form, a symbol of harmony between humans, nature, and the God, integrating mathematical reasoning with cultural and spiritual values.

The findings of this study also indicate that the lingko lodok rice-field structure contains various mathematical concepts that are highly relevant to the school mathematics curriculum. Figure 1 illustrates the relationship between the lingko lodok field pattern and several geometric shapes and concepts. In general, the arrangement of lines and regions within the lingko lodok field reflects the concepts of points, lines, circles, radius, arcs, sectors, triangles, and trapezoids.

The farming activities of the Wae Nakeng and Manggarai communities generally consist of four major stages: (1) the preparation of seedlings and land, (2) the planting stage, (3) the crop maintenance stage, and (4) the harvesting and post-harvest stage.

Farming in the lingko lodok fields of the Wae Nakeng community begins with preparing rice seedlings and opening or plowing the land designated for the nursery. Typically, the nursery plots are rectangular, with dimensions varying depending on the total size of the rice fields to be planted. After preparing the nursery, farmers sow the seeds evenly. The seedlings are then allowed to grow for approximately 21 days before being transplanted across the entire agricultural area. During the growth period of the rice seedlings, approximately when the seedlings are 2 weeks old, farmers begin plowing, constructing the rice-field embankments, and preparing the main fields so that the land is ready for planting within the following week (around the seedlings’ third week).

During the seedling and land preparation stage, activities such as determining the size of the nursery plot, plowing the field, and constructing the rice-field embankments represent concrete manifestations of the ethnomathematical activities of counting, measuring, and designing. The number of rice seedlings to be prepared is calculated based on its proportion to the total area of the land to be cultivated. This reflects counting activities in determining the quantity of seeds needed for a given land size. Similarly, farmers determine the size of the nursery plot using a specific proportion of the total agricultural land area, illustrating the activity of measuring. Furthermore, designing activity is clearly visible in the construction of the nursery plot, which is generally shaped as a quadrilateral (a trapezoid or a rectangle) and in the formation of straight-line embankments surrounding the field.

These practices demonstrate an intuitive understanding of mathematical concepts such as geometry, ratio, and informal measurement. In these processes, farmers implicitly apply concepts of length, width, area, and proportional reasoning when adjusting the size of the nursery plot and the number of seedlings to be planted according to the size of the land. Specifically, both the nursery-plot area and the quantity of seedlings exhibit a pattern of direct proportion, meaning that any increase or decrease in land area is followed by a consistent and linear change in these two variables. For instance, if a farmer cultivates a field with an area of k m^², he may require m kg of seedlings that must first be prepared in a nursery plot with an area of n m^². If the cultivated field area increases, the size of the nursery plot and the number of seedlings needed will increase proportionally, showing a direct linear relationship among these quantities. The proportional relationships between the number of seedlings (NS) and nursery-plot area (ANP) with respect to land area (AL) can be expressed as:

and

These ratios indicate that both the number of seedlings and the size of the nursery plot can be systematically determined by maintaining their proportional relationship to the land area. Thus, if the farmer cultivates another field of p m2, the estimated number of seedlings and nursery-plot area can be calculated using the direct proportion formulas:

kg and m²

After preparing the seeds and rice fields, the second step is to plant the rice seeds throughout the rice fields, known as rede. This activity typically involves multiple workers, with the number depending on the size of the field, the estimated completion time, and the financial resources available. For example: a respondent with ¼-hectare field generally requires about four workers to complete the rede process in a single day.

The planting stage (rede) reflects the application of counting, measuring, and locating activities. Farmers carry out planting after the seedlings are approximately 21 days old. During this stage, they consider the number of workers needed, the estimated working time, and the required budget, which is demonstrating the presence of counting in the rede process. Measuring is evident when farmers estimate the spacing between seedlings, which is typically around sa baka (equal to the width of a handspan). Locating is also reflected in how farmers position each seedling. In this case, the seedlings are not planted randomly but arranged with careful attention to spacing and alignment to form straight and orderly rows of rice plants.

In connection with mathematics education, various school mathematical concepts can be identified in the rede stage. Counting the age of seedlings can be used to teach addition and subtraction. Determining the number of workers needed based on land size and estimated work duration also illustrates the application of ratio and proportion concepts. Interview with a farmer revealed that a ¼-hectare field requires about four workers to complete rede within a single day. This indicates that the relationship among land area (AL), number of workers (x), and working time (t) can be expressed proportionally as follows.

This proportion illustrates that land area is directly proportional to both the number of workers and the working time, while working time is inversely proportional to the number of workers. Using this proportional relationship, the estimated working time (t’) can be calculated when the land area (AL) and the number of workers (𝑥′) are known.

For example, if the land area (AL') is 2 hectares and 32 workers (x') are available, the estimated working time (t') is:

Similarly, if the land area (AL') and the expected working time (t') are known, the required number of workers (x') can be calculated. For example, if the land area is ¼ hectare and the task must be completed in 2 days, then the required number of workers can be calculated as follow.

The determination of planting distance between seedlings also reflects the application of interval, distance, and consistent repetition of patterns across the field. This corresponds to the mathematical idea of an arithmetic sequence, where the equal planting distance represents the common difference (d) and the position of each planting point corresponds to the sequence’s terms (Un). Students can be asked to calculate the length or width of a side of a rice-field plot when given information about planting distance and the number of planting points. In such cases, the length (𝑝) of a side can be calculated using the formula for the 𝑛-th term of an arithmetic sequence (Un).

For instance, if one side of a plot contains 19 planting points with an equal spacing (d) of sa baka (approximately 15 cm), and the distance from the plot boundary to the outermost point is also sa baka (see Figure 2), then the length of that side equals to the 21-th term of an arithmetic sequence. It is from the 19 planting points and two points of the plot boundary.

Therefore, the length of the side can be calculated as follow.

p = U21

p = U1+(n – 1)d

p = 0 + (21 – 1)15

p = 300 cm = 3 m

After the rede activity, the rice field is maintained through irrigation, fertilization, and the removal of weeds (tawi). Approximately 3 - 4 weeks after planting, farmers conduct tawi, a weeding activity aimed at clearing unwanted plants that may hinder rice growth and development. One respondent shared that for a field of about 1 hectare, tawi is typically carried out by up to 12 workers and completed within 3 days. After tawi, the field continues to be maintained through fertilization, pest control, and sufficient irrigation.

Activities such as weeding (tawi), irrigation, spraying, and fertilization reveal the presence of counting, measuring, and explaining activities. Counting is evident when farmers calculate the age of the rice plants to determine the proper timing for tawi, fertilization, and spraying. Decisions regarding the amount and type of fertilizer or pesticide also involve counting. Farmers perform irrigation periodically at fixed time intervals, reflecting counting activities related to scheduling and frequency. Measuring is visible in how farmers determine fertilizer doses, the volume of pesticide mixtures, and the quantity of water needed for irrigation. These activities demonstrate practical understanding of measurement, volume, and water flow. Furthermore, explaining emerges through farmers’ reasoning when estimating the amount of water required for optimal plant growth based on field size and weather conditions.

Several school mathematics concepts can be found in crop maintenance activities. Estimating rice-plant age for tawi illustrates the use of arithmetic operations. For example, if farmers perform tawi on day m and rede was carried out on day n, the plant age can be calculated as u = m – n. The ideal age for tawi is generally 14–21 days or depends on weed growth. Thus, the date for tawi can also be determined using the following linear inequality.

Determining fertilizer quantities shows understanding of direct proportion between fertilizer needs and land size. If a farmer needs a kg of fertilizer for b hectares, the ratio is a : b. In addition, mixing pesticide with water before application reflects measurement of volume, concentration ratios, and arithmetic operations.

Approximately 2 to 2½ months after tawi, the rice plants are ready to be harvested (ako). The ako activity involves cutting the rice stalks and placing them into sacks. In Manggarai traditional culture, lepo (woven from pandan leaves) was used in place of modern sacks. After harvesting, the next process is rontok, which is the separation of rice grains from their stalks. This is followed by teter (winnowing) and finally drying the rice (pari) until it is ready to be milled. For a 1-hectare field owned by one respondent, the harvesting (ako) and rontok activities were typically completed in one day with 25 female workers and 7 male workers.

The harvesting (ako) and post-harvest stages (such as rontok, teter, and pari) contain strong ethnomathematical dimensions. Activities involving counting, measuring, locating, and explaining appear throughout the harvesting and post-harvest processes. During these stages, farmers consider the age of the rice plants, the number of workers needed, estimated working time, and the required budget. These considerations reflect counting activities. Farmers also store harvested rice in sacks (lepo) of various sizes and perform post-harvest processes (such as teter and pari) using containers of different capacities (roto, doku, and loce). These practices demonstrate measuring, involving mass and volume measurement using various containers. Harvesting also requires locating, such as precisely positioning the sickle to cut the rice. Furthermore, locating becomes evident during teter woja, when farmers hold the roto or doku above their head at specific angles (slowly moving from 0° to 90°) so that rice grains fall while the wind separates them from husks, dust, and empty chaff. In the harvesting and post-harvest stages, farmers base their decisions on clear reasoning and explanations. For instance, when determining the right harvesting time, farmers not only calculate plant age but also assess rice maturity based on grain color and fullness.

In relation to school mathematics, the harvesting and post-harvest stages encompass several mathematical concepts, including linear models/functions, geometry, and geometric transformations. A linear model with three variables emerges when estimating the daily cost of harvesting. Farmers need to estimate the budget required for the harvest based on the number of male workers (x), female workers (y), and the daily consumption cost (z). The daily wage for male and female workers is typically predetermined. In the Wae Nakeng community, the daily wage for men is IDR 100,000 and for women is IDR 75,000. Thus, the total daily cost can be modeled as:

During teter, the concept of angles becomes particularly apparent. The position of the roto or doku above the farmer’s head forms various angles as it moves from 0° to 90°. The angle between the container and the farmer’s head forms an acute angle (see Figure 3), which enables effective separation of the rice grains.

Furthermore, the teter context can be used to teach students how to calculate arc length. If a farmer uses a roto with radius r and moves it through an angle θ radians, the arc length p traced by the roto is:

In the agricultural practices of the Wae Nakeng and Manggarai communities, a variety of traditional tools and measurement systems are used. The traditional farming tools consist of the machete (kope), shovel (bancik), sickle (kandong), and containers like roto, doku, and lepo. These tools embody ethnomathematical elements embedded in local farming traditions. Generally, the kope is used to cut wood and branches on agricultural land. In the context of rice-field farming, it is also used to clean the bunds. The shovel (bancik) is used to build bunds, turn over the soil, and level the rice field after plowing. The sickle (kandong) is used for cutting grass and harvesting rice.

To support farming activities, the Wae Nakeng and Manggarai community also uses several traditional containers, including the basket (roto), winnowing tray (doku), and sack (lepo). The roto is used for storing harvested rice and for estimating its weight. It is made from bamboo that is about one year old (still relatively young), split into strips measuring between 1 and 1.5 cm. The doku is used for winnowing the rice to separate empty grains after threshing. The lepo is used to store and count harvested rice, and it is available in various sizes such as lepo warang, lepo wega, lepo telu warang, and lepo beka. Other traditional measurement units include tongka and re cie. The relationships among these units are summarized in the following Table 1.

The use of various traditional tools and measurement systems in the rice-farming practices of the Wae Nakeng and Manggarai community reflects a close relationship with mathematical knowledge that has developed naturally within their cultural traditions. These tool and measurement unit represent several universal mathematical activities such as counting, measuring, designing, and locating. Counting and measuring are evident in the use of the tongka unit and different lepo sizes to estimate and compare harvest quantities. Designing appears in the crafting of tools such as the kope, kandong, bancik, roto, and doku, which are made with careful consideration of form and function. The use of these tools also requires precise positioning to achieve optimal results, reflecting the activity of locating. For instance, when cleaning rice-field bunds with the shovel, the farmer must position the shovel vertically to produce a straight and neat bund.

The relationship between traditional tools and measurement systems in the rice-farming activities and school mathematics concepts is clearly visible. The shapes of the tools represent various plane and solid geometric forms. For example, the shape of kope can be used to introduce a composite geometric figure combining two right trapezoid (see Figure 4a). The bancik, on the other hand, resembles a combination of an isosceles trapezoid and a rectangular (see Figure 4b).

Moreover, the circular shape of the doku (see Figure 5a) incorporates numerous mathematical concepts such as radius, circumference, arc length, and area of a circle. Meanwhile, the roto represents the three-dimensional form of an open cylinder (see Figure 5b).

(b) an open cylinder in the shape of roto

The use of non-standard measurement units is also reflected in these traditional systems. Units such as tongka can be used to help students understand informal mass measurement before converting to standard units such as grams or kilograms. For instance, students can learn that one tongka of rice is roughly equivalent to 3–5 kg, allowing them to determine mass in standard units. Additionally, mathematical concepts such as arithmetic and algebra are embedded in systems like tongka and lepo, which illustrate ideas of multiples, ratios, and proportionality. The relationships among these units demonstrate multiplication, division, and unit conversion, aligning with foundational concepts in school algebra.