Learning and Teaching angles of Geometry

Learning and Teaching angles of Geometry

Introduction

To effectively teach mathematics, mastery of the subject matter, understanding of the reasoning of students and mathematical concepts is required, and knowledge of the different strategies of instruction(Ball &McDiarmid,1990; Fauskanger,2015; Harrington,1999). The conceptual approach rather than an instrumental approach should be used by teachers in answering students’ questions(Borko & Putman,1996; Tchoshanov,2011). The conceptual approach focuses on  “understanding the meaning of mathematical representation, explaining the reason why certain algorithms and procedures work in particular situations and establishing the connections between mathematical concepts instead of only facts and procedures.” However, the master of subject matter is not sufficient for one to be an effective teacher (Shulman,1996). Teachers need to have a clear understanding of what they know and how they teach(David & Simmt,2006; Mason &Davis,2013). In-depth knowledge of the topics students have difficulty grasping is required and the strategies needed to mitigate these challenges because a teacher’s knowledge affects student’s conception(Tirosh,200).

One of the key components of  the mathematics curriculum is Geometry(Common Core State Standards[CCSSI],2010;Ministry of National Education[MoNE],2013;National council of Mathematics[NCTM],2000).This is because ” geometry is a key element to understand and to facilitate students’ visualization and reasoning capabilities”(Clements & Battista,1992). The National Council of Mathematics emphasizes the importance of “analyzing characteristics and properties of two and three-dimensional shapes and developing mathematic arguments about geometric relationships. “This paper will focus on angles which are a subtopic under Geometry. The concepts to be covered are the definition of angles, the history of measuring angles, and how angles have been measured over time, the different ways in which angles are measured, methods for teaching angles, the challenges of teaching angles and the misconceptions of angles students and teachers may encounter.

Definition

An angle can be defined as the figure formed when two rays extend from the same point P. The size of the angle changes according to the rotation of the two arms around P. This can occur in the clockwise or counterclockwise

History of measurement of angles

The division of a circle is the method used in measuring angles. The measurement is based on degrees and arc length. A degree is “1/360 of the circumference of a circle. Angles are classified according to their measure. An exactly 90^{o angle} is referred to as a right angle. An angle less than 90^o is an acute angle. An angle greater than 90^o is an obtuse angle.180^o is referred to as a straight angle. An angle greater than 180^o is a reflex angle. An interesting question that often arises is why the measure of angles is up to 360^o. The people in ancient Babylonia created this system of measurement using the Sexagesimal system, which is to the base of 60 instead of the decimal system, which is to the base of 10. “They knew that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle into 360. With a base of 60,6 times, 60 was a natural choice. Furthermore, this coincided with their knowledge of astronomy of the last century BC: The year was divided into six equal parts, each having sixty-plus some more fractional number days.”

The modern way of measuring angles is the use of a protractor. “A protractor is a tool transparent plastic tool. It is usually semi-circular but also available in a full 360^o version. It has a small hole near the bottom of the protractor called the origin, which is aligned over the vertex of the angle to be measured.”

How to teach angles

According to Martinez, José Manuel & Ramirez, Laura. (2018), the teaching of students about angles the following four stages has to be followed. The first is exploring the concept of angles, how they can be defined and represented. The second is the Comparison and classification of angles based on attributes such as size. The third is the measurement of angle by degrees. The fourth is the observation of measurements and deduction of rules relating to the angle of a triangle.

The following is a teaching procedure on the measurement of angles based on the Common Core State Standard for Mathematics.

The student should be able to identify a “one-degree angle” as an angle that turns through 1/360 of a circle. This will form the basis for determining the measurement of angles.

The student will be required to cut out the “one-degree angle” above.

The objective of the lesson is recognition of angles as geometric shapes that are formed from the intersection of two rays sharing a common point and measurement of angles using a protractor. Looking at the circle above ray a and ray b share the same point of intersection, Point O. They form an angle the two rays being the sides. It turns 1/360 of the circle and is called a “one-degree angle.”

The student will be required to cut out the “one-degree angle” above. This will be used to draw angles with different measurements. The procedure will be; select an everyday piece of paper, the center of the paper should be labeled as point O, Ray a and ray b should be drawn terminating at the center of the circle.

To draw an angle using a protractor and ruler, you begin by drawing a ray using a ruler. Put a dot at the end of the ray, making sure it is the center of the protractor. The zero marking on the protractor should be on the ray. To measure an angle of 70 degrees, make sure that the dot on the paper is at the forty-degree mark. The protractor should then be removed, a line drawn to join the ray and the dot for the 70 degrees.

Misconceptions of angles students and teachers may encounter.

The following are the main mistakes students make when answering questions on angles — poor visualization. The students focus on the physical appearance for drawn figures and interpret it as a whole. For example, a student may perceive the first figure to have a greater angle than the second figure just because it has longer sides, yet it’s not true.

The second mistake is poor analysis. The student may have an understanding of the properties of angles, but they analyze them independently. The mistake students make about learning about angles is that they prefer cramming instead of understanding the concepts on angles.

There are three components of teacher knowledge of a subject, “Subject matter knowledge, pedagogical content knowledge, generic pedagogical knowledge.” Possessions of the above elements are vital for a teacher to be considered effective. Some teachers have not mastered the concepts relating to angles such as visually distinguishing right, obtuse, acute, and reflex angle; this, in turn, affects student understanding of these concepts. Finally, some teachers have trouble communicating the same ideas to students.

References

Martínez, José Manuel & Ramírez, Laura. (2018). Angling for Students’ Mathematical Agency. Teaching Children Mathematics. 24. 424. 10.5951/teacchilmath.24.7.0424.

National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

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Fauskanger, J. (2015). Challenges in measuring teachers’ knowledge. Educational Studies in Mathematics, 90(1), 57-73.

Borko, H., & Putnam, R. (1996). Learning to teach. In D. Berliner & R. Calfee (Eds.), Handbook of educational psychology (pp. 673–708). New York: Macmillan.

Tchoshanov, M. A. (2011). Relationship between teacher knowledge of concepts and connections, teaching practice, and student achievement in middle grades mathematics. Educational Studies in Mathematics, 76(2), 141-164.

Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4-14.

Davis, B., & Simmt, E. (2006). Mathematics for teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, 61(3), 293-319.

Mason, J., & Davis, B. (2013). The ımportance of teachers’ mathematical awareness for ın-the-moment pedagogy. Canadian Journal of Science, Mathematics and Technology Education, 13(2), 182–197. doi:10.1080/14926156.2013.784830

Tirosh, D. (2000). Enhancing prospective teachers’ knowledge of children’s conceptions: The case of division of fractions. Journal for Research in Mathematics Education, 5-25

Clements, D. H. & Battista, M. T. (1992). Geometry and spatial reasoning. In D. Grows (Eds.), Handbook of research on mathematics teaching and learning, (pp. 420- 464). New York: MacMillan.