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

Front 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.

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

Metal Frame of a Door

When Lochen walks around Lombokstraat in Utrecht, he interests in a house which has a nice design of metal frame on the door of it as shown in the picture below.

Calculate the number of quadrilateral in the metal frame!

Big Clock

Have you ever seen a big clock of a church in Lombokstraat? Someday, Thomas walks around and takes pictures of the clock, as seen as bellow.

Find the measure of the smallest angle formed by the two hands of clock!

Bridge

When Sarah walks around Lombok, she finds a bridge and takes a photo of it.

While standing in front of the bridge, she confuses whether the bridge is higher than the building behind it. Can you help Sarah? Explain your answer!

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

Since the second discussion within this article is talking about the role of models within RME, thus it much relates to the second characteristics of RME above. It is the use of models or bridging by vertical instruments.

Models within RME have different manifestation since it necessarily indicates essential aspects of mathematical concepts and structures that are relevant for the problem situation. It means that models within this theory can be served as materials, visual sketches, paradigmatic situations, schemes, diagrams, and even symbols in mathematics (Treffers and Goffree, 1985; Treffers 1987, 1991; Gravemeijer 1994). For instance, paradigmatic situation of repeated subtraction can be functioned as model that gives access to the formal long division algorithm.

The suitability of “models” to support the learning process accomplishes at least two important characteristics. First, they must be imaginable context or be originated in realistic. Second, it provides sufficient flexibility to be applied on more advance level. In brief, it means that a model acts as scaffolding which support the progression in vertical mathematizing without blocking the way back to the sources of original strategy.

In term of bridging role between informal and formal level, Streefland (1985) present that models can contribute to level rising by shifting from a “model of” to a “model for”. In brief, the “model of” is a model which represent close connection to the problem situation, while the “model for” is used to facilitate the organization of related and new problem situations as well as to reason mathematically. Freudhental (1970) also published the different between the two models above in his writings. He marked that “model of” is after-image of a piece of given reality, while “model for” is pre-image for a piece to be created reality.

In conclusion, the use of “models” in the implementation of RME at school should be observed carefully since it supports the progression in both horizontal and vertical mathematizing.

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.

Conditions:

• m ∠APC = m ∠BQC = m ∠ACB
• Sa, Sb, and Sc are squares on BC, AC, and AB respectively
• |AB| = c, |BC| = a, |CA| = b, and |PC| = d

Problems

1. Prove that : area Sb = area Rb and area Sa = area Ra
2. Prove that ab = cd

Solutions

1.  The proof is as follows

2.   The proof is as follows

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

Illustration

The proof of Ceva’s Theorem is as follows

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.

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

Proof 2 (Using Similar Triangles)

Let ABC is a right triangle with right angle at C as shown on the figure below.

We draw the altitude from point C, and call D its intersection with the side AB.

The proof of the Pythagorean Theorem is as follows

Euclid was eager to place this result on his book, but he came up with another way to prove the Pythagorean Theorem. One speculation is that the proof by similar triangles involved a theory of proportions, a topic not discussed until later in the Elements. Furthermore, the theory of proportions needed further development at that time.

Proof 3 (Euclid’s Proof)

The following is a summation of the proof by Euclid that can be found in Book I of Euclid’s Elements specifically in Proposition 47 and 48.

Proposition: In right- angled triangle, the square on the hypotenuse is equal to the sum of the squares on the legs.

Euclid begin with the Pythagorean configuration shown in figure below.

Then, he constructed a perpendicular line from B to the segment HI on the square of the hypotenuse. Both points J and K are the intersection of the perpendicular line and the sides of the square on the hypotenuse. The line also lies along the altitude to the right triangle ABC.

Furthermore, Euclid showed that the area of rectangle CHKJ is equal to the area of square on side BC and that the area of rectangle JKIA is equal to area of square on side AB. To prove it, he constructed segment BI and CF, to form the triangles AEC and AIB

The proof is as follows.

Similarly, it can be shown that rectangle CHJK must have the same area as square BCGF = BC × BC = BC². Then, adding two results, AB² + BC² = (AI × AJ) + (JK × JC). Since AI = AB², then we have AB² + BC² = (AI × AJ) + (AI × JC) = AI × (AJ + JC) = AI × AC = AC × AC = AC². So, in conclusion, we obtain that AB² + BC²= AC².

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.

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

Level 1. Analysis

Students at this level begin to analyze the properties of geometric figures. Through observation and experimentation, for instance, a student might declare “A square has 4 equal sides and 4 equal angles. Its diagonals are congruent, perpendicular, and bisect each other”. Even though students are able to list all properties of given geometric figures, they do not tolerate the properties to overlap because each property is separate from others. For example, they still insist that “a square” different with “a rectangle” because they may propose extraneous properties to bear such beliefs, such as rectangle has one pair of sides longer than the other pair of sides. As a result, interconnections between figures are still not discerned and the definitions are not yet appreciated.

Level 2. Abstraction (Informal Deduction)

At this stage, students can ascertain the interconnections of properties within figures as well as among figures itself. Learners, for example, begin to recognize that all squares are rectangle, but not all rectangles are squares based on their comprehension of the properties of each. Furthermore, both meaningful definitions and informal arguments of geometric concept can be generated to rationalize their reasoning. Even if formal proofs can be grasped, however, they do not see how the logical order could be altered nor do they see how different premises could be starting construction of a proof.

Level 3. Deduction

Either axiomatic system or the role of undefined terms, axioms, postulates, definitions, theorems, and proof is well understood. A learner at this stage is able to construct, not just memorize, geometric proofs and understand their meaning. Students at this level, however, judge that axioms and definitions are fixed rather than arbitrary. Therefore, they cannot yet consider non-Euclidian geometry because they geometric ideas are still accepted as objects in the Euclidian plane.

Level 4. Rigor

This geometry level is well known as the level of mathematician. Students at this level understand that the object of thought is deductive geometric systems, for which they associate axiomatic systems. In addition, learners can study non-Euclidian geometries with fully understanding.

This post will be continued with issue “properties of the levels” as soon as possible.

Reference:

Crowley, Mary L. “The van Hiele Model of the Development of Geometric Thought.” In Learning and Teaching Geometry, K-12, 1987 Yearbook of the National Council of Teachers of Mathematics, edited by Mary Montgomery Lindquist, pp.1-16. Reston, Va.: National Council of Teachers of Mathematics, 1987.

Mason, Marguerite. “The van Hiele Levels of Geometric Understanding”. Virginia: McDougal Little Inc

Before further discussion to answer the question arises, to begin with, I would like to describe how he understands the term of comparing fractions with like denominators. Spontaneously, the idea of utilizing “bar modeling” and “number line” came to my mind and it worked.

In the term of “bar modeling”, I introduce two equal bars in size which are representing two fractions that being compared. Let’s say I want to compare  1/4 and 3/4 . The first bar models the first fraction 1/4, while the second bar be 3/4 as shown in the illustration below.

It is obvious from the illustration above that  1/4 < 3/4

Furthermore, considering “number line” as the alternative, I asked him to put fractions (1/4, 2/4, 3/4 and 4/4) on the given number line in the correct order.

After that, I have him compare which one is bigger 1/4 or 3/4 and without any hesitation, he positively answered that  1/4 <  3/4. In addition, he concluded that the larger fraction is the one with the greater numerator as long as the fractions have like denominators.

In the other hand, different situation happened when he struggled with unlike denominators. I started to explain about equivalent fractions and common denominators but he was confused with it. Curiously, I checked his math textbook and found “multiply-cross trick” without giving any prior explanation about equivalent fractions or common denominators. Then, spontaneously, I asked for myself “Does the trick really make sense for third grader?” and my answer is “I don’t think so”.

Before I elaborate the best way to teach this material, in a brief, here I show you the rule of “multiply-cross trick” that I found on the textbook:

1. Multiply the numerator of the first fraction by the denominator of the second fraction and write down the answer under the first fraction.
2. Multiply the numerator of the second fraction by the denominator of the first fraction and write down the answer under the second fraction.

For example, suppose you are asked to compare these two fractions 4/7 and 3/9

By applying the rule, then you obtain

Since 36 is greater than 21, then the fraction 4/7 is greater than 3/9.

This trick, perhaps, is going to be easy for my brother or others third graders as long as they understand about multiplication. However, the idea of presenting this trick in his textbook disturbs my mind since it fails in providing any REASONING to understand the concepts.

So, what should we do to scaffold that idea such that it become more makes sense?”

(The answer of that question will be continued on this post as soon as possible)