Understanding Measurement: A Child’s Journey in Math
Item 3. Magnitude and Extent
The Educational Challenge of Measurement
“The educational challenge will be to find teaching situations that allow meaning construction of the essential concepts of measurement, for which the student will be involved, which should provide the necessary tools to function in his life as a citizen” (Chamorro, 2003).
1. Genesis of the Idea of Size and Measure in the Child
1.1. Steps to Overcome for the Child to Start Working with Size and Measure
- Consideration and perception of a magnitude.
- Conservation of magnitude.
- Regulation on the magnitude.
- Correspondence between numbers and quantities of magnitude.
Consideration and Perception of a Magnitude
The child must consider the properties of the object or collection of objects that are presented and thus differentiate and distinguish their subsequent isolation. The property is treated in the rest of the properties or attributes that they may have.
Steps to Overcome for the Child to Start Working with Magnitude
Conservation of Magnitude
The child has to identify what changes can cause changes in the object with the resulting variation in the extent treated as well as those to be left invariant.
At the time that students have acquired the idea that, if the object changes position, shape, size, or other property, however, one thing remains constant: that something is precisely this magnitude over which we want the child to be conservative.
Regulation on the Magnitude
The properties that define the magnitudes allow sorting; naturally, objects are treated. When the child exceeds the stages of consideration and perception of the magnitude and the preservation of it, will be able to establish relationships between objects and comparisons of the type “rather than” or “less than.” The possibility of ordering is intrinsic to the notion of magnitude.
Steps to Overcome for the Child to Start Working with the Magnitude
Correspondence Between Numbers and Quantities of Magnitude
The last step or stage to stress corresponds to the ability to measure itself. The fact of a comparison between objects and their subsequent management invites us to consider how intense is “more than or less than” the relationship. We say that an object weighs twice as much as another, three more times, etc.
As regards the construction of the notion of measurement, Piagetian studies indicate that the child must pass the following stages:
- Compare direct perceptual
- Moving objects
- Operation of the transitive property: indirect comparisons.
Steps to Overcome for the Child to Start Working with the Measure
Direct Perceptual Comparison
The child compares perception so that objects are presented and does not use any common measure or displacement. Only if direct perception does not give enough information, use intermediate objects composed of certain parts of your body (hands or feet if the length), but as mere support for the perception.
Move Objects
At this stage, the child finds the need to compare objects and moving them close enough to extract perceptual information, and if this approach cannot be performed, intermediate objects are used beyond his own body.
Operation of the Transitive Property: Indirect Comparisons
With the comparisons made in previous stages, the child feels able to perform arguments such as:
“If a = b and b = c then a = c”
Where the element b would be the intermediary for the comparison. This stage is linked to the conservation of quantities since they are treated from transformations (displacements and strains) thus demonstrating the conservation of those.
Operation of the Transitive Property: Indirect Comparisons
It is necessary and important to verify the transitive property as students in often it is the visual memory which acts as a support for this property.
It is worth noting the differences in the use of the transitive property we can be found in the different magnitudes, for example, several bands of different lengths. It is possible to arrange them in a ladder. (Visual comparison / sort in whole without explicit use of the transitive property) the mass of objects with a double pan balance. The display of the total order is not possible, be observable in this case, repeated and unnecessary heavy in our students, demonstrating the need to acquire the transitive comparisons.
1.2. Developmental Stages of Development of the Concept of Unity
It is important to note that, in the beginning, the agent used in the comparison of objects does not match the pattern or standard unit of measurement is often used.
Accuracy will only convince the student of the need for a standard or standard unit. Once the child has reached the operation of the transitive property that once the measure is established, it develops the notion of unity, whose constitution is the next evolution:
Theoretically, setting the application is as a secure unit. From an educational standpoint, the student aware of the importance and the need to set the drive is a key aspect of great importance that it requires special treatment by designing teaching situations that allow discovering the role played by the unit in the establishment of the measurement of magnitudes (Chamorro, 2003).
We can distinguish five steps in the formation of the unity of a magnitude:
- Lack of unity
- Object-Unit
- Situation Unit
- Figural unit
- Unit itself
Lack of Unity
The first approach to the measure is a strong perception. Thus, two objects are compared directly with each other, but this strategy quickly shows its shortcomings when it is the presence of a third object.
EXAMPLE: If the child is confronted with the comparison of three containers in the case of scale capacity, you can compare its contents without even using a measuring unit.
We Can Distinguish Five Steps in the Formation of the Unity of a Magnitude
Object-Unit
At this stage, the child considers the unit of measurement associated with the object itself. Strategies are frequently found in the unit of measurement are the constituent parts of the object itself to be measured.
EXAMPLE: To measure the ability of a liquid, which offer different smaller containers is common for children to use one whose form is more similar to the container whose contents are measured.
Situation Unit
At this stage, the unit of measure still depends on the object to be measured but is changed to other objects based on the relationship between them. To measure small objects and small units are used to measure large objects are used higher than the previous units.
We Can Distinguish Five Steps in the Formation of the Unity of a Magnitude
Figural Unit
The unit at this stage is losing the relationship with the object to be measured, although it is still associated with specific figures.
Such strategies are observed in the magnitude capability where, for example, the child has several units to measure any object, which amount to a true system of units of that size, maintaining the trend of measuring large objects large units and small objects with small units.
Unit Itself
When the unit manages to break free of the shape of the figure, the size and the object itself to be measured, is when you get the construction of the concept of unit of measure, the same for all figures or objects.
The unit of magnitude is not nothing but several particular magnitudes, but that is not associated with a specific figure. The first patterns emerge from the parts of the body (anthropometric measures) and, of course, the use of these units can yield good results when the same individual who measured, but the need for a uniform unit of measure is the agreement for establishing the system of measurement known as the Metric System.
2. Study of Linear Parameter: Length, Weight, and Capacity
2.1. The Scale Length in Children
Perhaps the length of magnitude more elemental constituyendo worked in education in most cases the intermediary as many others, assuming therefore an obstacle in their construction.
2.1.1. Size and Distance
When dealing with a volume perceptible objects, which are called objects filled, the length scale is based on its hardware. The distance, however, refers to the empty space between two objects, thus being different treatment of these situations.
The two notions are complementary, but the child may encounter difficulty in bringing each other.
To Get to Build an Effective Manner the Notion of Distance, the Child Must Develop Three Basic Conclusions (Belmonte, 2005):
- Conservation of the distance between two objects. When you set the distance between two objects, this is maintained even get in the way objects between them.
- Symmetry in the distance: the distance between A and B coincides with the distance between B and A.
- Inequality in the distance: to bring an object C, placed between A and B, the distance between A and C or between B and C is less than the distance between A and B.
Steps to Overcome for the Child to Get the Conservation of the Distance
Piaget says up to 7 years these properties are not consolidated.
2.1.2. Conservation of Length
There are three aspects to take into account the difficulties encountered by the student to isolate the length, position changes, changes of form, and the decomposition / recomposition.
- Position changes. Children may not keep the equality of two lengths when one of them suffers a displacement: Exclusive fixation on the endpoint / endpoint.
- Changes in shape. The child tends to make judgments based on issues not critical to the evaluation of the lengths as to the position of the ends, the number of curves, or the number of segments: It favors the straight lines are privileged segment number
2.2. The Magnitude of Mass in Children
The mass scale can be considered as one of the largest magnitudes of sensory perception and therefore has a marked error.
From the physical point of view has to tell the difference between mass and weight because they are different magnitudes. But this should not make us overlook the complementarity of these magnitudes since the weight of objects is what allows us to appreciate the mass of these.
There Are Two Main Aspects to Take into Account the Difficulties Encountered by the Student to Isolate the Mass Scale: the Volume and the Decomposition / Recomposition
- Volume. It is common perceptually ordered the mass of the objects according to their volume. And it has to take care with this aspect because if you use empty objects the student may be considered as in nothing since it weighs.
- Decomposition / recomposition. As in the case of length, the student can make erroneous judgments about the conservation of mass of an object after being decomposed and recomposed. Thus, if we decompose a lump of clay in various pieces, may be understood that the resulting mass is not the same.
2.3. The Scale Capacity in Children
The scale capacity is often regarded as equivalent to the size volume, but are not. Physically no different, but their mathematical models are very different: the capacity is a linear scale and volume trilinear magnitude.
As the Main Aspect to Highlight in the Acquisition of This Magnitude Are: the Form and the Decomposition / Recomposition
- The shape. To two differently shaped containers, it is common to assess the ability of the height. With children of certain ages, and although the liquid transfer is done in its presence, the visual perception of the level prevailing on the amount of liquid.
- Decomposition / recomposition. As in the case of other quantities, the student can make erroneous judgments about the conservation of the ability of an object after being decomposed and recomposed. So, if you give out the contents of containers and other containers that the child can understand that the amount of resultant liquid is no longer the same.
3.3. Legal Systems. Metric System
| SYMBOL | UNIT | SIZE |
|---|---|---|
| M | meter | length |
| kg | kilogram | MAS |
| s | second | TIEMPO |
| l | liter | CAPACIDAD |
3. Treatment Resources: The Problem of the Measure
It begins with a set of objects, and they’re going to highlight one of its measurable attributes that will allow the mathematical construction of a quantity.
The senses are going to provide information regarding the assessment of these properties or attributes, and thus can partition the set of objects treated. Each of these partitions (equivalence classes in mathematical terms) is what is called a quantity of magnitude.
Each magnitude number is formed by a set of equivalent.
If we take two objects with different mass quantities, we can say that one is heavier than another and establish its management.
The didactic transposition of the measurement of magnitudes is characterized, among other things, the existence of a variety of terms and vocabulary using floating means interchangeably both stocks and mathematical concepts and different social nature.
3.1. Shortcomings and Problems in the Teaching-Learning and Action Figures
- Inability of students to distinguish different magnitudes.
- Change of units. Order of magnitude.
The measurement is almost always fictitious and with a strong ostensive, which aims to replace the actual measurement of concrete objects (concrete objects as expressed by several and a unit that supports most of the activities proposed in class.)
That is why the notions of approximation, estimation, and management of a magnitude not often treated in the classroom.
- Ignorance of the usual methods of measurement.
- Numeracy measure.
- Vocabulary.
- Dialectic exact / approximate measure.
- Role of the errors.
“There is therefore a clear substitution of knowledge in which the real problems of measurement are replaced by arithmetic problems, the measurement processes by the use of formulas, conversions and exercises, which occupy more than half the working time spent to the extent they are an exercise in decimal.” (Chamorro, 2003, p. 229)
3.2. Tip of Progress in the Treatment of Action. Classification Process and Sequence. Materials for Teaching and Action Figures
The work for the construction of quantities is a progression made up of different classification and management activities, where possible teaching-learning sequence could be:
For the progression of these activities will be necessary in the classroom objects sufficient in number and variety, to provide rich contexts for students to encourage the establishment of relationships and comparisons, thus facing the student, as far as possible, difficulties involving the isolation of the magnitude
- Length. By overlaying the ends of two bands, one end indicates which of the two is longer. Only when no direct comparison possible (For example: is it a closet door?) Will need to use intermediate measures as noted for the measured length is smaller and notch (cord used) to make the comparison in the other object to be measured or use spans far superimposed one after another.
See examples in which evidence for a direct comparison of measurement: - Capacity. When used to transfer liquid from one container to other decisions based on the height of the containers are often sources of error.
- Masa. When performing a weighing of objects in the hands or dual scale pan, the dish that informs us falls far heavier.
- Time. When we made comparisons at this scale is very difficult to ignore the subjectivity of the events happen. It requires a high capacity for reasoning and deduction in this regard. Somehow, if we match the start of two events, it is easy to see which of them sooner.
Materials for Teaching and Action Figures
- MASS: double pan balance, sand.
- LENGTH: rope or flexible material, rigid band, tape, etc.
- CAPACITY: water or other liquid, sand, containers of different shapes and size, measuring cylinders, etc.
- TIME: hourglass, stopwatch, timer tape, etc.
- SURFACE: tracing paper, graph paper, scissors, tangram, etc.
- VOLUME: policubos, solids to join, etc.