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Contextual Math Problems Based on Field Trip to Lombokstraat , Utrecht, Netherlands

Decorative Lamp

In the middle of Lombokstraat, Anton sees a decorative lamp, as shown in the pictures below.

Front view

Front view

Front Side View

Front Side View

He is curious about the length of the wire used to form the lamps. If the distance between each lamp is 10 cm, calculate the length of the wire!

Dutch House

Dina takes a vacation in the Netherlands and visits Lombokstraat in Utrecht. She is curious about the height of a Dutch house with a funnel.

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Estimate the height of the house from the ground to the top of the funnel based on the picture above!

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Reflection Paper: The Didactical Use of Models in Realistic Mathematics Education (Part 1)

Referring to an article “The Didactical Use of Models in Realistic Mathematics Education: An Example from A Longitudinal Trajectory on Percentage” written by Marja van Den Heuvel-Panhuizen, I would like to share my understanding about general theory of RME and the role of models within this theory.

Realistic Mathematics Education (RME) conceptualizes Freudenthal’s (1971) thought of “mathematics as a human activity”. It means that mathematics is an activity of organizing problems from real situation or mathematical matter – which is called mathematization– rather than just the body of mathematical knowledge. Treffers (1978, 1987), furthermore, distinguished the perspective of mathematizing into “horizontal” and “vertical” mathematizing. Mathematical tools, in term of horizontal mathematizing, are used to organize or solve a problem related to daily life. In the other hand, vertical mathematizing indicates for all kind of re-organization and operation within the mathematical system itself.

Regarding to mathematizing, the characteristics of RME is also historically related to the levels of Van Hiele theory and Freudenthal’s didactical phenomenology. As a result, those theory compose the five characteristics (tenets) of RME (de Lange, 1987; Gravemeijer, 1994) which is generally described as follows.

1. The use of contexts in phenomenological exploration.

The starting point of mathematics learning in RME should use mathematics concept appearing in reality, such that pupils become immediately connected in contextual situation.

2. The use of models or bridging by vertical instruments

In this case, the term of “model” refers to situational and mathematical models developed by pupils themselves.

3. The use of pupils own creations and contributions

Using students’ own constructions and productions. Students’ own constructions and productions are meaningful for them, and so should be used as an essential part of instruction

4. The interactive character of the teaching process or interactivity

The interactivity principle of RME signifies that learning mathematics is not only an individual activity but also a social activity. Therefore, RME favors whole-class discussions and group work offering students opportunities to share their strategies and inventions with others. In this way, students can get ideas for improving their strategies. Moreover, interaction evokes reflection that enables students to reach a higher level of understanding.

5. The intertwining of various learning strands or units

The intertwinement principle means mathematical content domains such as number, geometry, measurement, and data handling are not considered as isolated curriculum chapters, but as heavily integrated. Students are offered rich problems in which they can use various mathematical tools and knowledge. This principle also applies within domains. For example, within the domain of number sense, mental arithmetic, estimation and algorithms are taught in close connection to each other

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Hypothetical Learning Trajectory (HLT) within the Three Phases of Design Research

Hypothetical learning trajectory (HLT) regarded as an elaboration of Freudhental’s thought experiment provides design and research instrument that is very effective for design research. The HLT supply a marked connection between an instruction theory and a real teaching experiment in class. It monitors how the instructional activities perform in natural way appropriate with the didactical activities planned by teacher.

HLT, a key part of instructional learning design, consists of three components: the learning goal, plan for the learning activities, and the hypothesis of learning process. Regarding those components, Simon (1995) refer the learning goal to the possibility of the learning proceed as to the direction which is predicted by teacher. Furthermore, the term of hypothetical learning process relate to teacher’s prediction of how the students’ thinking and understanding will progress during learning activities.

In terms of design research, Bakker (2004) describe that HLT has different function depending on each phase as follows

  1. During preliminary design, HLT is formulated as to guidance of a sequence of instructional activities involving several conjectures of students’ thinking that had to be adapted.
  2. During teaching experiment, HLT serves principles of what the teacher or researcher want to focus on in teaching, interviewing, and observing. It is possible that the HLT of the instructional activity during this phase needs to be adjusted for the next lesson. Several conditions that might caused the minor changes in HLT are anticipations that have not come true, strategies that have not been foreseen, activities that were too difficult, etc. However, it is very essential for teacher or researcher to report those kinds of changes supported by theoretical considerations.
  3. During retrospective analysis, HLT is used as guideline in determining what teacher or researcher should focus on the analysis. The researcher can compare those anticipations with the observation during concrete learning in class. As the result, the HLT can be formed after the retrospective analysis in term of providing a guideline for the next design phase.

To sum up, HLT is very useful in term of design and research instrument since it propose comprehensible step of teaching activities within all phases of design research. Moreover, the HLT helps teacher to reflect their teaching whether it needs any improvement or not. Therefore, the proceeding of HLT cannot be done in one design because it considers the feedback of each lesson as well as the implementation of its improvement in the next lesson.

References

Bakker, A. (2004). Design Research in Statistics Education. On Symbolizing and Computer Tools. Amersfoort: Wilco Press.

Simon, M. A. (1995). Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, 114-145.


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Ceva’s Theorem

This theorem is named after Giovanni Ceva (1647 – 1743), who proved it in his work De lineis rectis in 1678. However, this theorem was proved much earlier by Yusuf Al-Mu’taman ibn Hud at eleventh-century.

Ceva’s theorem states that given triangle ABC, and points D, E, and F along the triangle’s side, lines AE, BF, and DC are concurrent if only if

Capture3

Illustration

Capture

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Pythagorean Theorem

The Pythagorean Theorem is named for the ancient Greek mathematician Pythagoras (569 B.C – 500 B.C). It states the area of the square built upon of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.

Capture1

Thus, according to the illustration above, the Pythagorean Theorem can be written algebraically as an equation

Q1 + Q2 = Q3

The statement revealed that Greeks used area as a fundamental concept, not the length. As consequence, we have square root the area to define the length.

The theorem has numerous proofs including geometric proofs and algebraic proofs with some dating back thousands of years. Some of the proofs are described below.

Proof 1 (Using Rearrangement)

To begin with, I have a large square that contain four identical triangles as shown in the figure 1. After that, I want to move all of the triangles to another identical square. As shown in the figure 2.

Capture2

Therefore, the white space within each of the two large squares must have equal area.

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Being Acquainted with the van Hiele Model of Geometric Thought

The van Hiele model, in mathematics education, is a theory involving levels of geometrical thinking that can be used to assess students’ abilities as well as lead instruction. This theory was completed simultaneously by Dina van Hiele-Geldof and Pieree van Hiele and was published in their doctoral dissertation at Utrecht University in 1957.

The model is made of five levels that describe how children’ thinking process to reason in geometry. The children will pass through these levels as they improve from merely recognizing geometric figures to being able to work with in a variety of axiomatic systems.

The levels of van Hiele are described as follows:

Level 0. Visualization

At this initial level, students are able to recognize geometric concepts by only their holistic appearance. Geometrical figures, in addition, are identified based on their visual prototypes rather than their components or attributes. Students, for example, recognize that squares in figure (a) are completely different with rectangles in figure (b) in theirs’ mind.

square and rectangle

A student at this level, however, would not distinguish that all of the figures above have right angle or that opposite sides are parallel. Continue reading