This paper examines two foundational topics in elementary mathematics education. The first section distinguishes standard units of measurement from non-standard units, using a real-world example of measuring a door in meters versus hand spans to illustrate why standard units are more reliable and universal. The second section applies the van Hiele theory of geometric thinking to Pre-K through grade 6 students, explaining Levels 0 through 2 — visualization, analysis, and abstraction — and proposing classroom activities that promote geometric understanding through shape recognition, paper folding, and exploration of shape properties.
A standard unit of measurement offers a point of reference by which items of weight, length, or capacity can be described. It is a quantifiable value that helps every individual understand the relationship between an object and its measurement. For instance, volume can be expressed in metrics such as gallons, ounces, and pints. A non-standard unit of measurement, on the other hand, is something that may fluctuate or change in terms of weight or length depending on who is measuring.
To illustrate the difference, a door was chosen as the object of measurement, as it is an item found in everyday life. Using standard units of measurement, the door was found to be equivalent to 2 meters, or 200 centimeters, in length. This standard measure was selected because it can be applied consistently using tools such as a tape measure. Using non-standard units of measurement, hand spans were used to determine the length of the door, which was found to be equivalent to 10 hand spans. This non-standard measure was chosen because it is simple and easy to apply without tools.
The key difference between the two types of measurement is that the standard unit is more reliable and constant than the non-standard unit. For example, a child with a smaller hand will have shorter hand spans and will therefore find the door to measure more than 10 hand spans. This demonstrates that non-standard units are neither reliable nor consistent across individuals. A standard unit of measurement, by contrast, is universal and will yield the same result of 2 meters regardless of who performs the measurement (Morin, 2018).
The van Hiele theory can be used to promote the understanding of geometry for Pre-K through grade 6 students. Levels 0–2 of the theory comprise visualization, analysis, and abstraction. Importantly, each level within the theory has its own symbols and language. To increase learning and understanding of geometry, students are advised to follow the hierarchical order of the levels.
Level 0 is visualization. At this level, students can use visual perception together with non-verbal thinking. They can recognize and identify figures by their overall shape and compare figures to objects they encounter in everyday life, classifying them accordingly. For instance, at this level students are able to determine whether a shape looks like a triangle or not.
Level 1 is analysis. At this level, students begin examining and naming the different features and properties of geometric shapes. They deepen their understanding of geometry through hands-on activities such as cutting and folding pieces of paper into shapes. For example, a student can cut a piece of paper and determine whether it is a square by checking whether all its sides are the same length.
Level 2 is abstraction. At this level, students are able to recognize the relationships between figures and their various properties. As a result, they become capable of constructing meaningful definitions, providing justifications for their reasoning, and drawing diagrams (Vojkuvkova, 2012).
"Three hands-on activities building geometric thinking skills"
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