**10. Discussion**

#### **10.1. Design phase**

and in 10% sugar solution time is 1.42 s, and so on. Therefore we can present the results as in

If we continued this work then we would obtain functional relationship between percentage of sugar solution and time taken by the ball sneaking in the sugar solution. So students can describe

**Figures 12** and **13** (personal collection of photographs).

**Figure 13.** Five percent and 10% sugar solutions with "time sneaking".

**Figure 12.** Five percent sugar solution with "time sneaking".

**Figure 11.** Sugar solutions with various concentrations.

20 Science Education - Research and New Technologies

There is one interesting thing that happens when the sugar solution reaches to 50% solution. It turns out that "ball clay" is not sinking, the ball in the solution is not dropped or immersed in the 50% sugar solution, but went up and floating. A member of the researcher who is a junior high school teacher was alarmed and shocked and thus raises the question "Why not down?" Why and why? She relayed the question over and over again while still in the physics lab, during a process of designing instructional materials that have not been brought into the junior secondary class.

Because the solution that is available only up to 40–50%, while 45% is not yet available, so he had the initiative and desire to deeply make a solution of 45% immediately and she wanted to know how the time crept to the 45% sugar solution. The research team soon made the 45% sugar solution, and measured how long time (how many seconds) a plasticine "clay ball" felt down in the sugar solution. In fact it took 61,22 seconds.

For an ideal situation after discussion with the team (persons of mathematics, physics, com‐ puter science, and mathematics teachers), the team suggested we should also know the dura‐ tion of time sneaking of the "ball" in the solutions of sugar 41, 42, 43, 44, 46, 47, 48, and 49%, but due to time constraints and opportunities, the team finally just gave a prediction of dura‐ tion time that in graph will look roughly like the image below in **Figure 15** (the graph is made by the researcher team using excel).

For junior secondary students, drawing graph smoothly was not a main target. There was no obligation for students to draw graph smoothly. But the researcher and developer team in this study try to interpret and predict the form of graph look like. It encourage students and recommend the researcher team to investigate further for the numbers around the 45%. It provides an impetus and a recommendation to investigate further around earlier numbers.

Equations or mathematical models in relation to the concentration of the solution with time of "sneaking ball" into a particular function, again for junior high school students, have not been the main target. The junior high school students are required to put or plot dots of various observation results as coordinates (solution, time) or coordinates (time, solution).

#### **10.2. Discussion in implementation phase**

Before getting into the observation using teaching materials (model) that have been pre‐ pared in the laboratory to the students, worksheets were also presented which aim to explore

**Figure 15.** Graph prediction of the sugar solutions.

Mathematics Instruction Based on Science Using Didactical Phenomenology Approach... http://dx.doi.org/10.5772/intechopen.68437 23

**Figure 16.** Incorrect graph {(1,5), (6,15)} graph.

down?" Why and why? She relayed the question over and over again while still in the physics lab, during a process of designing instructional materials that have not been brought into the

Because the solution that is available only up to 40–50%, while 45% is not yet available, so he had the initiative and desire to deeply make a solution of 45% immediately and she wanted to know how the time crept to the 45% sugar solution. The research team soon made the 45% sugar solution, and measured how long time (how many seconds) a plasticine "clay ball" felt

For an ideal situation after discussion with the team (persons of mathematics, physics, com‐ puter science, and mathematics teachers), the team suggested we should also know the dura‐ tion of time sneaking of the "ball" in the solutions of sugar 41, 42, 43, 44, 46, 47, 48, and 49%, but due to time constraints and opportunities, the team finally just gave a prediction of dura‐ tion time that in graph will look roughly like the image below in **Figure 15** (the graph is made

For junior secondary students, drawing graph smoothly was not a main target. There was no obligation for students to draw graph smoothly. But the researcher and developer team in this study try to interpret and predict the form of graph look like. It encourage students and recommend the researcher team to investigate further for the numbers around the 45%. It provides an impetus and a recommendation to investigate further around earlier numbers. Equations or mathematical models in relation to the concentration of the solution with time of "sneaking ball" into a particular function, again for junior high school students, have not been the main target. The junior high school students are required to put or plot dots of various

Before getting into the observation using teaching materials (model) that have been pre‐ pared in the laboratory to the students, worksheets were also presented which aim to explore

observation results as coordinates (solution, time) or coordinates (time, solution).

junior secondary class.

22 Science Education - Research and New Technologies

by the researcher team using excel).

**10.2. Discussion in implementation phase**

**Figure 15.** Graph prediction of the sugar solutions.

down in the sugar solution. In fact it took 61,22 seconds.

knowledge of whether the student has been able to plot the points of known coordinates. In the "worksheet" the known point (1.5) and point (6.15), appears from the existing worksheets, students were asked to plot them in the Cartesian coordinates. There is a group of students who describes a straight line that contains the point (1.5) and (6.15), that is supposed to only two points, namely (1.5) and (6.15). The students plotted correctly, but wrote (0,5) wrongly, it supposed to be (1,5) (see **Figure 16**). Students should only plot the coordinates (1,5) and (6,15), not necessary to draw the line from (1,5) to (6,15) (the graph were made by students and were photographed by the authors, **Figure 16**‐**18**).

In plotting the coordinate points of {(1,5), (2,7),(3,9),(4,11),(5,13) (6,15)}, generally students worked corretly, but some of students were not correct. The graph of the points are dots as figuring out by students in the **Figure 19**, not as a line segment as figuring out in the **Figure 16** and **20**. (the graphs of **Figure 19** and **20** were made by the students and photographed by researcher team).

Teachers began to deliver lessons after asking a number of questions above and confirmed that the correct point coordinates are the pairs of points (*x*, *y*) such that *x* and *y* are integers in a couple of points {(1.5),(6.15)} and do not represent a straight line.

**Figure 17.** {(1,5), (6,15)} correct graph.

**Figure 18.** {(1,5), (6, 15)} correct graph.

However, the research team did not worry, because in general the students were able to plot the coordinate points that should be described in the coordinate plane.

#### **10.3. Sugar solutions and graph**

The next steps, after the students were able to draw coordinate points, they start to learn the part of sugar solutions in relation to viscosity (velocity) of sneaking ball in the various sugar solutions (percent solution of 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50%). They are exposed to the tools that have been recorded in the power point presentation. Furthermore, students use stopwatch (in their mobile phone) to measure the sneaking time of "ball clay" in each of the sugar solution. In this case the student does not measure speed, but measures how long it takes for sneaking "ball clay."

Some observations of groups of students are outlined in the following table:


**Figure 19.** {(1,5),(2,7), (3,9) (4,11), (5,13), (6,15)}.

Mathematics Instruction Based on Science Using Didactical Phenomenology Approach... http://dx.doi.org/10.5772/intechopen.68437 25

**Figure 20.** {(*x*,*y*) *y* = 2*x* + 3, *x* dan *y* real dengan 1 ≤ *x* ≤ 6}.

However, the research team did not worry, because in general the students were able to plot

The next steps, after the students were able to draw coordinate points, they start to learn the part of sugar solutions in relation to viscosity (velocity) of sneaking ball in the various sugar solutions (percent solution of 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50%). They are exposed to the tools that have been recorded in the power point presentation. Furthermore, students use stopwatch (in their mobile phone) to measure the sneaking time of "ball clay" in each of the sugar solution. In this case the student does not measure speed, but measures how long it takes for sneaking "ball clay."

Solution 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% Time 1.22 1.55 1.93 2.45 2.85 4.76 5.28 6.0 6.95 61.22 ~

the coordinate points that should be described in the coordinate plane.

Some observations of groups of students are outlined in the following table:

**10.3. Sugar solutions and graph**

**Figure 19.** {(1,5),(2,7), (3,9) (4,11), (5,13), (6,15)}.

**Figure 18.** {(1,5), (6, 15)} correct graph.

24 Science Education - Research and New Technologies

By plotting the points in the above table, there was obtained the following graph (**Figure 21**).

The graph in **Figure 21** indicated the original students' work that can be plotted using excel without any adjustment, except for the data of 35% just predicted. But after entering the data corrected by 35% and the estimate (approximate) data for a solution of 41, 42, 43, and 44%, the graph time‐solution was obtained as follows:

**Figure 22** represents the relationship between percentage of sugar solution and the sedimen‐ tation time of the ball. Each point in the graph represented the number ordered of *solution* (*%*) *and time*.

Students either individually or in groups in the classroom have already understood the con‐ cept of the functions and relationships. Modeling of the sugar solution above depicts a phe‐ nomenon that in order to precipitate an object into the solution, the more concentrated the solution the longer the time required to precipitate an object. In other words, the more concen‐ trated the solution is, the greater the obstacles encountered objects to penetrate the solution (see **Figures 23**–**25** as students' work after their observation the solution‐time. The tables are made by the students, but photographs are made by the researcher team as the personal collections).

**Figure 21.** Plotting graph of sugar solution to sedimentation time.

**Figure 22.** Plotting the percentage of sugar solution versus sedimentation time of plasticine ball.

The conclusion above can be used as a metaphor for our body liquid. If the liquid of our blood in our body more concentrated, the more difficult this liquid transforming objects (e.g., blood carries oxygen" from the heart to the brain). If transport is hindered then the patient will feel pain in his head. **Figure 26** is a circulation system of our body, the blood from the heart transports the oxygen to the brain. When the blood concentration gets high, then the stability of our health will influence.

About the extent to which the student can give reasons why the following arrow diagram is a function and why is not function, descriptions of student work is displayed as follows:

Students above (**Figure 27**) understood the relationship, as they wrote "A Member of A is only be paired with one quantity," even though, in fact, the relationship is "a very simple relation‐ ship among two sets," as far as the two sets are associated. In a particular association, this relationship was named function, this group gives reason "special relationship in A which paired with exactly one member of C." Suppose "special relationships that map each member of A with exactly one member of C" (see **Figure 27**).

Students (or other groups) (**Figure 28**) state as to which they answer the following, "because it is the relationship between the set‐1 and set‐2."

However, the function according to researcher, the term written by students has not written correctly. It is supposed to be the function "The special relationship that links each element in the set‐1 (domain) with exactly one element in the set‐2 (codomain)" **Figure 29**.

"(a) is a relation which is a function from A to B," while

"(c) is the relation which is not a function, because one of the members in A has two images in B."


**Figure 23.** Students' work of Cohort‐1.

Mathematics Instruction Based on Science Using Didactical Phenomenology Approach... http://dx.doi.org/10.5772/intechopen.68437 27

**Figure 24.** Students' work of Cohort‐2 with a correction by researcher.


**Figure 25.** Students' work of Cohort‐3.

The conclusion above can be used as a metaphor for our body liquid. If the liquid of our blood in our body more concentrated, the more difficult this liquid transforming objects (e.g., blood carries oxygen" from the heart to the brain). If transport is hindered then the patient will feel pain in his head. **Figure 26** is a circulation system of our body, the blood from the heart transports the oxygen to the brain. When the blood concentration gets high, then the stability

**Figure 22.** Plotting the percentage of sugar solution versus sedimentation time of plasticine ball.

About the extent to which the student can give reasons why the following arrow diagram is a function and why is not function, descriptions of student work is displayed as follows:

Students above (**Figure 27**) understood the relationship, as they wrote "A Member of A is only be paired with one quantity," even though, in fact, the relationship is "a very simple relation‐ ship among two sets," as far as the two sets are associated. In a particular association, this relationship was named function, this group gives reason "special relationship in A which paired with exactly one member of C." Suppose "special relationships that map each member

Students (or other groups) (**Figure 28**) state as to which they answer the following, "because

However, the function according to researcher, the term written by students has not written correctly. It is supposed to be the function "The special relationship that links each element in the set‐1 (domain) with exactly one element in the set‐2 (codomain)"

"(c) is the relation which is not a function, because one of the members in A has two images

of our health will influence.

26 Science Education - Research and New Technologies

**Figure 29**.

in B."

**Figure 23.** Students' work of Cohort‐1.

of A with exactly one member of C" (see **Figure 27**).

it is the relationship between the set‐1 and set‐2."

"(a) is a relation which is a function from A to B," while

**Figure 29** is a problem to be asked for students whether this diagram is a function or not and why? Similarly, **Figure 30** is a question for the students, whether this diagram is a function or not? And why? Whereas **Figure 31** is student's reasons why the diagram is a relation and function or why the diagram is a relation but not a function. (**Figures 29**–**31** are personal collection of photographs.)

Although the proposed language is less precise, at least the students have an idea that they can distinguish between the functions and relationships. While other students understand the word "function" as a means "to" or "benefit," as the answer to the following students:

**Figure 26.** The circulatory system [53].

**Figure 27.** Students' work of function definition.

**Figure 28.** Students' work.

**Figure 29.** Relation as function.

Mathematics Instruction Based on Science Using Didactical Phenomenology Approach... http://dx.doi.org/10.5772/intechopen.68437 29

**Figure 30.** Relation not as function.

**Figure 31.** Reason for a function and not a function.

**Figure 32.** Students' work and reason why this relation is not function.

**Figure 29.** Relation as function.

**Figure 28.** Students' work.

**Figure 27.** Students' work of function definition.

28 Science Education - Research and New Technologies

"There is one member of A that has more than one image in C, then it is not called a function from A to C" (see **Figure 32**, personal collection of photographs).

What do the specific things on the phenomena of "fireworks","sugar solution" and "faucet" are in fact a function that each phenomenon can be expressed as diagram of arrow, or with a table, or with the ordered pairs, or with the chart coordinates.

Although the students' understanding of the concept of relationship is less perfect, the students have already understood the concept of function. They observed a sugar solution and sedimen‐ tation time of plasticine ball in each solution through a power point presentation. They detected them using stopwatch provided (or using their own Hand phone or mobile phone (HP)).

#### **11. Conclusion**

From the study of the implementation of mathematics instruction based on science using didactical phenomenology approach we conclude that (1) the prototype of learning math‐ ematics can be made simply and scientifically in laboratories either by using sophisticated equipment or by using a simple way, as long as all the equipments can produce two groups of quantities, (2) the implementation process of mathematics learning in the classroom does not always use original tools such as original equipment set in the laboratory, but use equipment or software or a power point presentation as a tool or medium for the presenta‐ tion of the photo or video animation of the laboratory equipment (video water discharge, video sugar solution, and "deposition" of plasticine ball, fireworks video, and video of swivel angle and discharge of water), (3) the reaction of students toward learning materials of mathematics based on science using didactical phenomenology shows positive attitudes and enthusiasm, (4) achievement of mathematical ability even though a group of students who study mathematics based on science using didactical phenomenology approach have a higher average than students who study using conventional approach, statistically both are not significantly different, (5) there are differences in improvement of mathematical ability among students who study mathematics based on science using didactical phenom‐ enology approach (0.48) with students who studied with conventional approach (0.36). But both are in the same categories (middle) and mathematical models were found to show stu‐ dents the results that can be interpreted. Model of fireworks is considered as linear, water discharge models are considered as linear, and model of the sugar solution is considered as a graph arch.

When the researcher team made an experiment in laboratory, there is an interesting find‐ ing. When the clay ball was put on the sugar solution, the smaller percentage of the value of sugar solution, the faster the rate of sedimentation of "clay ball" and the higher con‐ centration of sugar solution or the more concentrated of sugar solution, the slower the clay ball penetrates the solution. So the time to reach the base of the bottle is getting long. When looking at the 40% sugar solution, the ball still can be awaited, when a solution to be 45% the ball still could be awaited although it requires longer time. However, when observing the 50% solution, a teacher who helped designing the study shows surprise and astonishment, "Why is this happening?" In fact she connects the question, "What to do with death?" Then our mutual discussions with the belief held by strengthening the teacher. Approximately what causes happen so? Yes, if the clay ball stops (or floats), it means the same as our blood in our body was stuck because of concentrated so that it can no longer carry oxygen. Interestingly, this teacher seems to associate a sugar solution with the body fluids or blood fluid in our body. The phenomenon of nature (physics) is that a severe type of ball clay is smaller than the density of the sugar solution, so that the "ball clay" floats. If the weight is of the same type, then the ball will be hovering in the 50% sugar solution. For clarifying this situation to the students, then the teachers shares readings about the relationships between viscosity of our blood and maintenance of our health.

"There is one member of A that has more than one image in C, then it is not called a function

What do the specific things on the phenomena of "fireworks","sugar solution" and "faucet" are in fact a function that each phenomenon can be expressed as diagram of arrow, or with a

Although the students' understanding of the concept of relationship is less perfect, the students have already understood the concept of function. They observed a sugar solution and sedimen‐ tation time of plasticine ball in each solution through a power point presentation. They detected them using stopwatch provided (or using their own Hand phone or mobile phone (HP)).

From the study of the implementation of mathematics instruction based on science using didactical phenomenology approach we conclude that (1) the prototype of learning math‐ ematics can be made simply and scientifically in laboratories either by using sophisticated equipment or by using a simple way, as long as all the equipments can produce two groups of quantities, (2) the implementation process of mathematics learning in the classroom does not always use original tools such as original equipment set in the laboratory, but use equipment or software or a power point presentation as a tool or medium for the presenta‐ tion of the photo or video animation of the laboratory equipment (video water discharge, video sugar solution, and "deposition" of plasticine ball, fireworks video, and video of swivel angle and discharge of water), (3) the reaction of students toward learning materials of mathematics based on science using didactical phenomenology shows positive attitudes and enthusiasm, (4) achievement of mathematical ability even though a group of students who study mathematics based on science using didactical phenomenology approach have a higher average than students who study using conventional approach, statistically both are not significantly different, (5) there are differences in improvement of mathematical ability among students who study mathematics based on science using didactical phenom‐ enology approach (0.48) with students who studied with conventional approach (0.36). But both are in the same categories (middle) and mathematical models were found to show stu‐ dents the results that can be interpreted. Model of fireworks is considered as linear, water discharge models are considered as linear, and model of the sugar solution is considered

When the researcher team made an experiment in laboratory, there is an interesting find‐ ing. When the clay ball was put on the sugar solution, the smaller percentage of the value of sugar solution, the faster the rate of sedimentation of "clay ball" and the higher con‐ centration of sugar solution or the more concentrated of sugar solution, the slower the clay ball penetrates the solution. So the time to reach the base of the bottle is getting long. When looking at the 40% sugar solution, the ball still can be awaited, when a solution to be 45% the ball still could be awaited although it requires longer time. However, when

from A to C" (see **Figure 32**, personal collection of photographs).

30 Science Education - Research and New Technologies

table, or with the ordered pairs, or with the chart coordinates.

**11. Conclusion**

as a graph arch.

Observing this phenomenon, our research team is interested in observing and making a math‐ ematical model of the graph. Apparently, the graph becomes asymptotically at 50% solution. In fact, it still needs to be investigated in a solution with a lower concentration, e.g., 49, 48, 47%, and so on, or fragments are more accurate as 49.5, 49, 48.5, 48, 47.5, 47, 46.5%, 46, 45.5%, and so on. We are talking to mathematicians and they advised to build mathematical models. For students, graphed predictions appear as in **Figure 15**.

Noting the benefits of mathematics instruction based on science using didactical phenome‐ nology approach and its consequences on the students, teachers, and on student achievement, the team delivered the following suggestions.

