# Which aspects of math are arbitrary

## Quantitative comparison of surface area with arbitrarily chosen unit dimensions. Can different surfaces have the same area?

### Table of Contents

1. Subject of the lesson

2. Learning objectives of the lesson

3. Position of the lesson in the teaching unit

4. Subject of the lesson

5. Objective of the lesson

6. Execution of the class situation

7. Content-related learning requirements

8. Factual information

9. Didactic reasons

10. Methodological justifications

11. Progress planning

12. Literature

13. Appendix

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I. Lesson Draft for Exam Lessons in Mathematics

Class: 3b (21 students, 9 girls and 12 boys)

Time: 8.40 a.m. - 9.25 a.m. (2nd hour)

Subject teacher: Ms. XXXX

Seminar leader: Ms. XXXX

### 1. Subject of the lesson

Introduction to the geometric size range of area. - Development of the concept of area by qualitative and quantitative size comparisons of areas.

### 2. Learning objectives of the lesson

Superordinate learning objective of the lesson:

By gaining concrete experience of qualitative and quantitative comparison of surfaces, the pupils should develop a concrete conceptual conception of the surface area and thus promote their spatial imagination.

In detail, the students should ...

- be able to distinguish precisely the terms line, area and area linguistically.

- know how to compare different surfaces directly with one another in terms of their surface area by cutting, assembling and laying on top of one another.

- Recognize the individuality of body measurements through the indirect comparison of areas with the help of non-standardized units of measurement and gain insight into the need for standardized units of measurement.

- learn to be able to break down an area into meaningful sub-figures (unit squares) (promoting figure-ground discrimination) and thus to determine the area.**- Understand the principle of area invariance by learning that differently limited areas as well as areas with different dimensions can have the same area.**

- learn to display different areas with the same area on the Geoboard by stretching rubber rings that form the outline of a surface.

- learn to be able to determine the area in the iconic representation of given figures with the help of the geoboard.

- Further develop and consolidate the various learning objectives of the lesson unit according to individual focus areas, so that their visual perception skills are further trained.

### 3. Position of the lesson in the teaching unit

(1) Development of ideas about the terms “area” and “area”.

(2) Qualitative comparison of surface area through disassembly and assembly. - "Does the carpet fit in the living room?"

(3) Measure with arbitrary units of measure. - Quantitative comparison of the surface area of everyday objects with the help of one's own body measurements.

(4) First ways of mathematically calculating the area with arbitrarily chosen unit dimensions.**(5) Quantitative comparison of the surface area with arbitrarily chosen unit dimensions - "Can different areas have the same area?"**

(6) Introduction of the geoboard

(7) Comparison of areas on the Geoboard

(8) Various exercises for the quantitative comparison of areas of figures (double lesson)

### 4. Subject of the lesson

Quantitative comparison of area with arbitrarily chosen unit dimensions - "Can different areas have the same area?"

### 5. Objective of the lesson

The pupils should understand the principle of area invariance by working with units of measurement by learning that differently limited areas as well as areas with different dimensions can have the same area.

### 6. Execution of the class situation

I've known class 2b since I started my apprenticeship in May 2004. Since the beginning of the 2004/2005 school year, I've been teaching mathematics for five hours a week on my own responsibility. The learning group consists of 9 girls and 12 boys. Of these 21 children, 13 are of foreign origin. With regard to the subject of mathematics, I cannot observe any significant differences in performance between children of German and non-German origin. However, this fact has a very clear effect in the linguistic area. As a result, there can be considerable differences in learning speed when working on tasks that require a certain understanding of the text. For this reason, I consider it necessary to visualize certain work assignments in the subject of mathematics (see 10 methodological reasons). At the same time, however, I instruct the students to gradually develop their skills in this area.

Overall, I see a motivated learning and working atmosphere in the class. Most students are open-minded and positive about mathematics. However, I experience the class as extremely lively. Some students find it difficult to adhere to agreed rules of conversation so that they speak into the class without being asked (e.g. XXXX, XXXX). Other students, on the other hand, have problems concentrating in conversation phases and following their classmates in silence and listening (e.g. XXXX, XXXX, XXXX). Overall, it can be observed that in phases of joint reflection and evaluation, the students still show little independence in order to be able to have a joint constructive conversation. For this reason, I still have to consciously set impulses and guide class discussions (see 10 methodological reasons)

In the last few months I have worked specifically with the students in this area and introduced some rituals in order to carry out the lesson with as little interference as possible. In addition, together with the students and Ms. XXXX, I have drawn up a class contract that contains basic rules of conduct. As an intervening measure in the event of repeated non-compliance with the class contract, I write the name of the student concerned on the board. At the beginning of the week I hang four little stars on the board. As soon as there are two names on the board, I remove a star. If there are still stars on the board at the end of the week, the class receives a reward in the form of a mental arithmetic game (“math soccer”). As a result, both the negative and the positive actions of the individual have a direct impact on the group.

Individual students, such as XXXX, XXXX, XXXX and XXXX, sometimes find it difficult to devote themselves to a task continuously and with concentration. They are then often noticed by preoccupation with other things and I have to repeatedly remind them of their work assignment. In the action-oriented and discovery phases of the lesson, this is not so strongly expressed (see 9 didactic reasons). I also observe a strong teacher focus in this class. This expresses itself in the form that numerous students ask for my help before they independently develop work orders or ask their table neighbors. In order to guide the students to more independence and to relieve myself of the teaching role, I introduced the bracket system in the class. This system enables me to help the students in a structured way. Other measures ("table of the month", "green and red points", etc.), which I cannot list in detail here, support effective and disruptive math lessons.

The students are familiar with the work and social forms that occur in the lesson. The students still have problems with the independent scheduling of their work steps. For this reason it is necessary that I give you various acoustic (bell) and symbolic (symbol cards) orientation aids (cf. 10. Methodical explanations).

The students' ability to perform and abstract as well as the pace of work are very different. Whereby this manifests itself in a different form in the area of geometry than in the area of arithmetic. There are students, such as XXXX and XXXX, who have considerable learning difficulties in arithmetic, but who, in contrast, can absorb content from geometry at an above-average rate. As a rule, I counter these differences through various qualitative and quantitative differentiation measures (see 10 methodological reasons). XXXX, XXXX, XXXX and XXXX have a particularly quick grasp of many mathematical areas, as they can quickly recognize relationships and transfer what is known to new content. In addition, they can give decisive impulses to the learning process of the class through their differentiated oral contributions.

### 7. Content-related learning requirements

In addition to the thematic content of areas, area and area invariance, the students have certain learning requirements that are important for this lesson:

**Competencies learned from previous teaching units:**

a) "Tangram" teaching unit (end of the 1st school year)

- Laying out different areas using geometric shapes (see 8th factual analysis) as well as adding figures shown on a reduced scale

- (Notional) decomposition of surfaces into sub-units (figure-ground discrimination)

- Getting to know basic properties and conceptual assignment of geometric shapes

b) Lesson with the topic "Axial symmetry" (end of the 2nd school year)

- Handling of geometric plates

- Creation of two congruent surfaces by folding, cutting, laying, mirroring and drawing (first experience of the principle of surface invariance)

- Development of several elements of the visual perception ability (see 9 didactic reasons)

- Dealing with methods of guessing and estimating

c) Teaching unit on the subject of lengths (middle of the 2nd school year)

- How to use the ruler**Competencies learned from this lesson**

- Linguistic differentiation between the terms area and area

- Ability to be able to qualitatively compare two different surfaces with one another in terms of their content.

- Realization that a uniform measure must be used for measuring and comparing the surface area

- Ability to determine the area of geometric surfaces with a unit square.

Overall, I have so far been able to observe that the students' insight into the principle of area invariance (see 8th factual analysis) is only limited.

As part of the teaching unit on the subject of axis symmetry, I was able to observe a relatively strong heterogeneity in the area of geometric thinking and imagination. According to Piaget's epistemology (cf. Lauter 1997), however, all students are at least at or above the level of symbolic-visual thinking (cf. 9 Didactic Justifications). There are a number of students who are already at the level of logically concrete thinking (including XXXX, XXXX, XXXX, XXXX).

### 8. Factual information

This lesson focuses on comparing different people *Surfaces* about measuring their respective *Area*. Through various exercises (see 10, methodological reasons), the students should learn the principle of *Area invariance* understand. In the following I would like to define these three mathematical terms in more detail.

Under *surface* one understands an arbitrarily curved or flat structure in space, in particular any boundary (surface) of a spatial figure (see Meyer Großes Taschenlexikon, p. 108).

*Areas* Like lengths and volumes, they belong to the geometric dimensions.

“The area of a flat figure is determined by the number of unit squares it contains. To determine the area of a surface F means that the surface F is a real number *m (F)* to be assigned (measurement function), which has the following properties:

1. *m (F)* is not negative,

2. *m (F1)* = *m (F2)*if F1 is congruent to F2,

3. *m (F)* = *m (F1)* + *m (F2)*if F is composed of F1 and F2.

[...] The unit of measurement for calculating the area is the square meter (abbreviation qm or m²). A square with a side length of 1 m has an area of 1 square meter. From the basic unit of 1 qm the following is derived: 1 km² [...] 1 ha [...] 1a [...] 1 dm² [...] 1 cm² [...] 1 mm² [...] " (Schülerduden 2004, p.256f).

From a didactic aspect (principle of isolating difficulties), comparisons with standardized units of measurement such as cm² or dm² as well as calculating an area using formalized equations are only the task and content of the school years in lower secondary level (see Radatz / Rickmeyer 1991, 70). At the center of the elementary school is the propaedeutic of area calculation, in which areas are compared directly and indirectly with one another (Radatz / Schipper 1983, 154).

Franke describes the principle of area invariance as follows:

“Two areas have the same area if they

a) are congruent, i.e. they can be laid one on top of the other in such a way that they cover each other exactly and nothing protrudes from any surface,

b) have the same decomposition, i.e. each of the surfaces can be broken down into the same partial figures or, conversely, composed of the same partial figures,

c) are designed in the same way, i.e. each figure can be laid out without gaps and without overlapping with the same number of unit areas (e.g. squares, triangles or hexagons) "(Franke 2000, 246).

In the context of this lesson, the focus is on the knowledge that two surfaces have the same area if they are designed in the same way.

**[...]**

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