Examination of geometric solids with special consideration of the cube: The students should get to know geometric solids and their properties. They should further develop their spatial imagination by working with geometric bodies on different levels of representation (enactive, iconic, symbolic).
Position of the lesson in the teaching unit:
Condition analysis
Situation of the learning group
Factual analysis
Didactic decisions
Teaching objective
Appendix
Bibliography
1. Introduction to the topic "Body" - Recognizing and assigning body shapes (1 hour)
The students should recognize the difference between "surfaces" and "bodies" by arranging body shapes and surfaces (everyday material) and describing their properties. Furthermore, the students should arrange the bodies among themselves (sphere, cube, cuboid, cylinder) according to their characteristics and get to know their characteristics and concepts. They should deepen their newly acquired knowledge by kneading a body of their choice.
2. Bodies and surfaces – drawing surface shapes (1 hour)
The students are to discover the connection between bodies and surfaces by drawing surface shapes with the help of packaging material.
3. Building with everyday material - making buildings from packaging material (1 hour)
The students should deepen their knowledge of the properties of bodies by building fantasy buildings from packaging material.
4. Free building with wooden cubes (1 hour)
The students are to use playful exercises with wooden cubes to build, recreates and describe fantasy buildings.
5. Building cube buildings according to pictorial specifications – determine numbers (1 hour)
The students are to determine the number of cubes in illustrations of cube buildings and recreates them for control.
6. Introduction of construction plans – exercises for the creation of construction plans (1 hour)
The students get to know the representation of the blueprint for a cube building. They should apply their newly acquired knowledge by finding and building as many different blueprints as possible for cube buildings with four cubes.
7. Building with the cube - inventing blueprints and building cube buildings (1 hour) The students should train their spatial imagination by drawing blueprints and building cube buildings.
8. Examine cube buildings – linking to multiplication tasks (2 hours)
The students should apply their knowledge by finding and presenting multiplication tasks to self-built cube buildings and pictorial representations.
Condition analysis
Situation of the learning group
In class 2, I have been supervised by my mentor in class since May. This supervised lesson comprises two hours per week.
Class 2 consists of 10 girls and 10 boys aged 7 to 9 years. According to Piaget, all students have thus reached the stage of concrete operations. This stage is characterized by the fact that thinking is bound to concrete ideas. At this stage of development, the act of thought is "compositional" and "reversible".1 With regard to Piaget's investigations into the development of geometric thinking, children of this age group can recognize perspective changes, see spatial layers and distinguish body shapes according to their characteristics.2
The oldest student in the class is xxx. He has been in the class since the beginning of the school year. xxx was previously taught in a language healing class and repeatedly the second grade. The Turkish student xxx speaks well german, but has e.g. T. Problems oral instructions sufficiently understand. The student xxx often finds it difficult to put her interests aside appropriately. When asked by the teacher to deal with the topic, she often reacts with withdrawal or refusal.
Basically, the general working attitude of the students in this class is good and the willingness and motivation to learn is great. The social climate in the learning group is characterized by a good sense of community, a high willingness to be tolerant and help towards weaker students.
In the learning group, large differences in performance can be observed with regard to mathematics.3 The high-performing children of the learning group stand out in a special way because they immediately recognize connections, have a quick task of understanding and contribute well to the lessons with qualitative statements and proposed solutions. Very weak students of the learning group stand out through the safe application of reproductive issues. Most students in the class show good average performance. The pace of learning and work corresponds for the most part to the performance of the learning group. Due to the strong performance gap, qualitative and quantitative differentiation measures are necessary.4
The selected topic will be dealt with in the classroom for the first time with this unit. The learning group has been familiar with the surface shapes triangle, rectangle, square and circle since the first year of school. The children of the learning group usually know the cube from games, the construction kit or as teaching material in random experiments. In the previous hours to this unit, the students, with special consideration of the EIS principle5, got to know the cube as a body shape next to ball, cuboid and cylinder. Furthermore, they can name the properties of the cube in their own words. In addition, the learning group got to know the presentation of the blueprint for a cube building and applied it on the basis of the first discovering exercises. The wooden cubes required for this lesson with the edge length of 2 cm are not yet known to the students; they have so far worked with cubes of the edge length 3.5 cm freely and purposefully. During the previous hours to this unit, the terms "above", "under", "next to", "right", "left", "in front", "behind" were repeated. In addition, the terms "cube", "sphere", "cuboid", "cylinder", "body", "surface", "corner", "edge", "blueprint" and "cube building" were introduced. Furthermore, the learning group is familiar with the social forms required for this lesson (chair circle situation, partner work, class lessons), organizational forms (chair circle, work at the place, frontal seating arrangement) and forms of action (free or bound teaching conversation, work on worksheets, work with material). With regard to the great motivation that comes with building and planning cube buildings, today's lesson suggests a successful work.
Factual analysis
In mathematics, the cube is a fixed, three-dimensional geometric figure bounded by six planes. It has eight corners, six faces and twelve edges. The surfaces are equal-sized, congruent squares. The edges are of the same length, with two surfaces colliding with each edge. At each corner, three surfaces and three edges always meet.6 The unrolling of a cube leads to the formation of cube nets. Each cube net consists of six equally large, contiguous squares.7
For the cube as a geometric body, there are different models; the solid model (compact body), the edge model and the surface model. With the cube as a solid model, cube buildings can be built. Basically, cube buildings are referred to as bodies that are composed of cubes of the same height in such a way that neighboring cubes with a square side surface touch each other fully.8
Example of a cube building
Abbildung in dieser Leseprobe nicht enthalten
In the translation of two-dimensional cube buildings into the three-dimensional perspective, images of side views can be a help. However, a cube building cannot always be clearly recre built on the basis of a side view, since an object can be represented from above, from the front, from behind as well as from the right or left. A cube building can only be clearly recreated if a blueprint is available that makes complete statements about the object. A building plan is defined as an "abstracted, graphically represented, two-dimensional representation of a spatial given"9. The footprint of a cube building can form the floor plan of a blueprint. Each floor plan is divided into squares, the digit of which indicates the number of cubes that are superimposed on each other in the corresponding fields.10
In principle, construction plans are considered identical if they can be converted into each other by rotation. In addition, cube buildings can be identical despite different construction plans by transferring them to other positions by rotation or tilting movements. A "Würfelzwilling" with a blueprint 2 can therefore be transferred by tilting movement into a cube building with the blueprint 1 1.11 Despite rotational and tilting movements, cube buildings with an equal number of cubes offer numerous combination possibilities, as the arrangement of the cubes as well as their standing surfaces can be varied. For cube buildings with eight cubes, 14 construction plans can already be created for a stand area of two squares without taking into account rotational and tilting movements.
Didactic decisions
According to the educational standards in mathematics for primary education, the pupils should have spatial imagination at the end of the fourth grade and be able to relate two- and three-dimensional representations of buildings to each other.12 In the present lesson and unit, these competences are initiated. In the Lower Saxony core curriculum mathematics for primary school, the planned teaching unit is the content-related competence area "Space and Form"13, especially the topic "body and plane figures"14 Associate. One of the expected competencies at the end of class two is that the students can name geometric bodies and surface shapes and their properties and recognize them in the environment. In addition, the planned lesson within the area of "Space and Form" belongs to the topic "Orientation in Space"15. At the end of the second school year, the students should be able to orient themselves in the room and describe situational relationships in the room and in the level in their own words. Furthermore, in the planned lesson, the construction of different cube buildings from pre-ed numbers of cubes will be linked to the topic of "surfaces and room contents"16 Instead of. The process-related competencies that are taken into account in today's hour are communicating/arguing and presenting/using didactic material.17
The school-internal material distribution plan, which is based on the textbook "Mathematics Primary School 2nd School Year"18 includes the topics "Getting to know basic geometric shapes", "Making models of bodies" and "Drawing and comparing surface shapes"19, which are integrated into the planned teaching unit. The construction and drawing of cube buildings and building plans is not addressed in the textbook. A further concretization of the requirements of the core curriculum by specialist conference decisions of the school does not yet exist.
A central task of mathematics lessons is to offer students an opportunity to acquire basic geometric knowledge and skills that enable them to participate in social life.20 These knowledge and skills should be important both in the current and in the future life situation of the students. This aspect of application orientation can be realized with regard to the selected topic.
The children of my study group move in a three-dimensional environment determined by geometric shapes, figures and bodies. This geometric structure is difficult to recognize or penetrate without the competences of a spatial idea; it forms the basis for the development of the predominantly spatial environment.21 In order to be able to orient themselves in this environment, the children need a good spatial imagination both at the present time (e.g. when crossing the street, during sports activities) and in the future (professional life). Furthermore, in our three-dimensional world, in which all possibilities of writing down, logging, photographing, etc. provide two-dimensional images, it is always necessary to imagine spatial objects or processes on the basis of two-dimensional images.22 This applies in particular to successful learning in all subjects (e.B illustrations in books). The examination of geometric bodies thus supports the students of my learning group in opening up their living environment and is also of fundamental importance for their cognitive and intellectual development.23
Especially in mathematics lessons, spatial imagination is crucial for number perceptions, number relationships and operations.24 The development of geometric thinking should help my learning group to internalize mathematical concepts and relationships and to deepen the insight into areas of arithmetic thinking. Furthermore, especially the children in my learning group, who have learning difficulties in the arithmetic area, can be taught a positive attitude to the subject of mathematics with the treatment of the selected topic. Furthermore, in the sense of interdisciplinary teaching,25 the spatial perception of the pupils is additionally promoted in sports, work and art lessons.
In the planned teaching unit, the methodical sequence of stages provided for by Radatz/Schipper, among others, is adhered to.26 The gradual development of a spatial imagination with special consideration of the linking of different levels of representation enables my learning group to gain experience about geometric bodies, about situational relationships and about the movement of bodies in space. In the planned unit, I focused on working with the cube, as this geometric form encourages the students to play and experiment and also fascinates with its simple, highly symmetrical shape.27 In addition, the learning group has already gained many previous experiences about this body from its everyday world, which can be linked to in this teaching unit. Furthermore, the cube as a didactic material offers important aids, especially for differentiated teaching.28
In order to develop the spatial imagination, the students should deal in today's hour both with dice acting and mentally with the spatial position of cubes (head geometry). Different levels of representation are linked together; the children recreate cube buildings with cubes on an enactive level on the basis of symbolic signs (building plans) (principle of internalization and interlocking of the representational levels)29. The students thus gain security in the alternation between the individual levels of representation and gradually reach a higher level of abstraction of the spatial conception.
The goal of communicating and arguing should be pursued with the partner work. In the sense of multi-channel learning30 different learning input channels are addressed in this task, as the students draw blueprints and look at cube buildings (visually), build with the cube (tactilely) and exchange their results with a partner (auditory). In the sense of stabilizing practice31 the learning group should deepen and consolidate the learned skills and abilities for building and representing a cube building with the help of a blueprint.
In this teaching unit, I limit myself to the massive model. In addition, I do not yet address the aspect of the identityity of blueprints by rotation, since the learning group deals with bodies for the first time (didactic reduction).
Teaching objective
Rough learning objective:
The students should train their spatial imagination by drawing blueprints and building cube buildings.
Fine learning objectives:
Technical objectives:
The students...
- reactivate and apply their knowledge to create a blueprint by drawing their own blueprints and building cube buildings according to a blueprint,
- solve their work task concretely (with cubes) according to drawing specifications (construction plans) by building cube buildings to building plans,
- Recognize space-location relationships by drawing blueprints for cube buildings or building cube buildings according to construction plans,
- Produce cube buildings in the imagination by operating with cubes mentally,
- further develop their creativity by inventing cube buildings,
- train their visuomotor coordination by building with cubes,
- know and use the principle of number decomposition by drawing blueprints for cube buildings with eight cubes,
- find ways to check their results,
- to find a suitable blueprint for iconic representations of cube buildings (symbolically) by assigning the corresponding blueprint to a cube building (quantitative differentiation),
- assign the corresponding, representational cube building to a building plan (symbolically) by comparing the building plan with their cube buildings,
- further develop their ability to communicate and argue by describing cube buildings with terms such as "in front", "behind", "above", "under", "next to", "left", "right" and using other introduced mathematical terms (corner, edge, surface) appropriately.
Procedural goals in the affective and social field:
The students...
- further develop their personal responsibility and independence by carrying out the work task responsibly,
- further develop their reliability by complying with agreements and rules during the independent work phase,
- train their ability to concentrate by listening to the candidate and each other,
- further develop their social skills by cooperating with each other during partner work,
- be encouraged in their ability to work in a team by working together,
- strengthen their self-confidence by naming or showing their classmates the building that matches the selected building plan.
[...]
1 cf. Zech, Friedrich: Grundkurs Mathematikdidaktik. Theoretische und praktische Anleitungen für das Lehren und Lernen von Mathematik. 10. Aufl. Weinheim und Basel: Beltz Verlag, 2002, p. 91.
2 cf. Radatz, Hendrik; Rickmeyer, Knut: Handbuch für den Geometrieunterricht an Grundschulen. Hannover: Schroedel Schulbuchverl., 1991. p. 12.
3 cf. Commented seating plan
4 cf. Methodological decisions
5 cf. Zech (2002), p. 104.
6 cf. Franke, Marianne: Didaktik der Geometrie in der Grundschule. 2. Aufl. München: Elsevier GmbH, 2007. p. 152.
7 Cf. ibid. p. 152.
8 cf. Reinke, Tanja: „Baupläne von Würfelgebäuden“. In: Grundschule Mathematik. Geometrie: Raumvorstellung. Nr.10, 3. Quartal, Seelze Velber: Friedrich Verlag, 2006. p. 14.
9 cf. http://de.wikipedia.org/wiki/Bauplan (6.05.2007).
10 cf. Radatz, Hendrik; Schipper, Wilhelm; Dröge, Rotraut; Ebeling, Astrid: Handbuch für den Mathematikunterricht. 2nd school year Hanover: Schroedel, 33- 43.
11 cf. Radatz/Schipper/Dröge/Ebeling (1998), p. 126.
12 cf. Kultusminister der Länder in der BRD (Hrsg.): Beschlüsse der Kultusministerkonferenz. Bildungsstandards im Fach Mathematik für den Primarbereich. Beschluss vom 15.10.2004. München: Luchterland 2005. p. 10.
13 Niedersächsisches Kultusministerium (Hrsg.): Kerncurriculum für die Grundschule. Schuljahrgänge 1-4. Mathematik. Unidruck Hannover, 2006. p. 26.
14 Ibid. P. 27
15 Ibid. P. 26
16 Ibid. P. 27
17 cf. Teaching objectives
18 Leppig, Manfred (Hrsg.): Mathematik Grundschule 2. Schuljahr. Berlin: Cornelsen Verlag, 1993.
19 Cf. ibid. PP. 72–74.
20 cf. Niedersächsisches Kultusministerium (2006), p. 7.
21 cf. Radatz/Schipper (1983), p. 141.
22 cf. Eichler, Klaus-Peter: „Räumliches Vorstellungsvermögen entwickeln“. In: Grundschule Mathematik. Geometrie: Raumvorstellung. Nr.10, 3. Quartal, Seelze Velber: Friedrich Verlag, 2006, p. 40.
23 cf. Radatz/Rickmeyer (1991), pp. 7-8/p. 34.
24 Cf. ibid. P. 8
25 cf. Niedersächsisches Kultusministerium (2006), p. 10.
26 cf. Radatz/Schipper/Dröge/Ebeling (1998), pp. 122-126.
27 Cf. ibid. P. 123
28 cf. Methodological decisions
29 cf. Zech (2002), pp. 116-117.
30 cf. - Gudjons, Herbert: Handlungsorientiert lehren und lernen. Schüleraktivierung – Selbsttätigkeit – Projektarbeit. Bad Heilbrunn/Obb.: Klinkhardt, 2001. p. 61.
31 cf. Zech (2002), p. 209.
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