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=  **__Understanding Periodic Motion Through Pendulums__**   =

**Goals of the Lesson:**
In this lesson students will be able to create linkage between geometry, algebra and physics concepts by first observing motion of a pendulum of varying length. Successful completion of this hands on experiment requires a clear understanding of the various academic concepts and terms, observation and recognition of periodic motion, data collection strategy, data plotting and analysis, and graphing. Data analysis will lead to comprehension of a square root function and operation amongst other mathematical operations.

**Academic Vocabulary:**

 * **Pendulum****:** A body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. (1)
 * **Periodic motion:** Motion that recurs over and over and the period of time required for each recurrence remains the same. (2)
 * **Period**: The time it takes for a pendulum to go from one extreme to the other and back.

Van Hiele Model of Learning:
In 1950s Dutch couples Pierre van Hiele and Dian van Hiele-Geldof theorized a cognitive theory of Geometric development which later became known as van Hiele model (Brahier, 2009). This Theory grounded in the assumptions that geometry is learned in sequential steps that are age independent. Learning takes place when students are progressed sequentially through five steps. Crowley (1987), explains the elements of this five step model as visualization, analysis, informal deduction, formal deduction, and rigor. In the visualization stage, shapes are identified with no clear understanding of the governing theories and relationships. In the next stage, analysis of the geometry begins and students become aware of the basic properties and attributes of a geometric figure; for example, recognizing that there are three sides and three angles in a triangle. Learners will be able to establish and comprehend the interrelationship between the properties in the next phase, the informal deduction stage. This understanding is further refined and is evident in the formal deduction stage, where theoretical proofs are understood and definitions become axiomatic. Learners are only then capable of comprehending abstract concepts in the last stage, the rigor. = =

**The Lesson Plan:**
In this lesson, the five steps of the Von Hiele Model are interpreted in a more abstract fashion, and used to guide the students from little to no understanding of a pendulum, Level 0, to a strong and abstract understanding of the mathematics involved, Level 4.

**Level 0: Visualization**
The teacher introduces the students to what a pendulum is and the vocabulary used throughout the lesson. Students are paired up, and each group is given a string, marked in 5cm segments, a weight and a stopwatch. Students are given a few minutes to simply play with the pendulum, getting a feel for how it works.
 * From playing with the pendulum, the students will observe that the pendulum motion is like the movement of a swing, but do not know the attributes of a pendulum.

**Level 1: Analysis**
Instruction is given on the first part of data collection. One student in each pair is to hold the pendulum at the 5cm mark. A second student times 30 seconds on the stop watch, while the first student counts the number of swings. The students then divide 30 by the number of swings to get the period or the length of time for one swing. This data is recorded on a chart.

Once all groups of students have finished collecting their first set of data, they are instructed to gather data holding the pendulum at 5cm increments through 50 cm, calculating the period of each set of data.
 * From the data collected, the students will see some trend in the number of swings through the different incremental lengths of the pendulum, but they do not have the skills to see linkages among the variables used in the experiment.

**Level 2: Informal Deduction**
Students are given the materials to create a poster and asked to graph their data, using the length of the pendulum (L) as their x value and the period (T) as the y value. The teacher leads a discussion on the shape of the graph, taking particular note of how the period is much more sensitive to a shorter pendulum length.

If a graphing calculator is available for students, a Power Regression can be performed on the data to introduce the students to the square root function. If not, the teacher uses a pre-made graph of the square root function to compare to the students' graphs.
 * At this level, the students will be able to observe from the graphs the relationship between the two variables (T) and (L) and will have a basic understanding of how this relationship links to one another.

**Level 3: Formal Deduction**
The teacher instructs the students that the generalized equation for their graphed function is T=k√L. Students are asked how they can determine the value for k, and a discussion is moderated by the teacher. Students then use their collected values for T and L to determine a value for k.
 * The students can now analyze and explain the various relationships of the variables with a more in-depth understanding of the relationships among the variables used in the equation. The students are also able to appreciate the usefulness of the square root formula.

**Level 4: Rigor**
Students are asked how they think a change in the weight at the end of the pendulum will affect the period values. After a poll of the class is taken, the students are asked to experiment in their pairs. After a period of time, the students are brought back together for a whole class discussion on their findings. (The weight at the end of the pendulum has no effect on the period. It is only altered by a change in the length of the pendulum.)
 * At this level, the students know, understand and have developed a better appreciation and reasoning pattern behind the relationships among algebra, physics and geometry concepts using the pendulum activity. Students are able to develop their own formula and can compare and analyze the different mathematical concepts and axiomatic systems.

**Benefits of the Van Hiele Model:**
What are the key success factors and benefits of applying the Van Hiele Model that maximize the learning potentials of students?

The elements of the five-step model of the Van Hiele model and its application has to be fully embraced and understood by the teacher. The language used by the teacher in explaining the mathematical task or experiment including the various mathematical concepts is key in the progression of students’ thinking and reasoning ability from one level to the next since learning, in this model, is sequential.

The mathematical task should be well-designed and supervised by the teacher, like the pendulum activity that the students will be instructed to do themselves. This exercise being hands-on gives the students the opportunity to observe, learn and make deductions from the results of the activity.

The Van Hiele model cuts across other learning theories, like the sociocultural theory, where the students’ zone of proximal development is challenged and enhanced. The students’ social interaction likewise allows the use of language as a tool for learning where they exchange, compare and help their peers in the analysis and formulation of all possible solutions to the problem.

The instructional design and learning experience becomes more meaningful with the use of this learning theory despite the students being at varying levels of reasoning. This learning model moves the teaching approach away from the “parrot math” process of problem-solving to one that involves a deeper and broader understanding of mathematical concepts and relationships.

Another equally important benefit of the use of this theory is that its successful implementation enhances the confidence of the students at learning other mathematical concepts and systems and move to doing more complex, abstract tasks and reasoning levels.

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**References:**
(1) [] (2) [] Brahier, Daniel J. (2009) Teaching Specific Mathematics Content. In Teaching Secondary and MIddle School Mathematics, 3rd Ed (pp. 213-214). Allyn & Bacon: Boston, MA. 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.

Malloy, Carol E. "Perimeter and Area through the Van Hiele Model." Mathematics Teaching in the Middle School 5, pp.87-90, 1999.

Mathematics Education Student Association, The University of Georgia. The Mathematics Education, Vol. 18 #1, 2008.