Visualization is extremely important for students learning and trying to understand math. Math is an abstract discipline. No one can "see" math but one can see how math can be used to real life. The problem with abstract concept is that each student may understand the abstract concepts under his / her past experience, and may not be the same as what is intended. Visualization allows the students and the teacher to have a common ground for further discourse. It is similar to making thinking visible in Geode.
Visualization via applications has been one of the most popular ways for teachers to introduce different math concepts. From counting apples to calculating height of buildings via trigonometry functions, or calculating time of arrival based on moving speed of a vehicle, etc. students can "see" how math principles can be used. This can be achieved through actual participation in solving a practical real life problem, or watching a video of a problem (such as Jasper), and then solving it, whether individually or collaboratively.
Another way for students to understand math is through graphics and animation on the computers. Sine waves, exponential functions, geometry, etc. can easily be manipulated using software such as the site indicated above. Visualization on the computers also allows a student to manipulate different variables to see the effect of the changes. Students can also observe "patterns" in such manipulations to deduce what changes and what stays the same, and come up with reasons for the behavior. An example of this is the Sum of Three Angles lesson under Angle and Parallel Lines in the Middle School (Geometry) section. As the student moves the triangle to the right or left, it is shown that the bottom left and right angles don’t change. Not only the “bigger” triangle doesn’t change, the “smaller” triangle preserves the same angles. The lesson Problem about Angles (1) in the same section introduces a problem to the student, and also provides a hint button to provide additional information. No description is provided but by manipulating the graphics, the student can see how the angles remain the same as one moves the red dot. The creative use of animation reduces significantly the amount of explanation that is required and this can be positive to a lot of students.
For students who are confident in deducing math principles, having the abilities to manipulate graphical representations and to see the effect of these changes instantaneously on the screen will be useful to them. However for students who may require extra help, such tools may seem cold and unfriendly. The hints provided in some lessons are useful, but in cases where the students still do not grasp the principles after exhaustion of the hints, additional help from the teachers or other students should be supplemented. Learning math does require one to process the concepts individually, but there are often cases where collaborative learning helps reinforce the student understanding and comprehension of the subject. The problem with many of the software visualization systems is that the students are left on their own in the learning, and it misses out on the collective engagement and knowledge sharing with peers.
Visualization via applications has been one of the most popular ways for teachers to introduce different math concepts. From counting apples to calculating height of buildings via trigonometry functions, or calculating time of arrival based on moving speed of a vehicle, etc. students can "see" how math principles can be used. This can be achieved through actual participation in solving a practical real life problem, or watching a video of a problem (such as Jasper), and then solving it, whether individually or collaboratively.
Another way for students to understand math is through graphics and animation on the computers. Sine waves, exponential functions, geometry, etc. can easily be manipulated using software such as the site indicated above. Visualization on the computers also allows a student to manipulate different variables to see the effect of the changes. Students can also observe "patterns" in such manipulations to deduce what changes and what stays the same, and come up with reasons for the behavior. An example of this is the Sum of Three Angles lesson under Angle and Parallel Lines in the Middle School (Geometry) section. As the student moves the triangle to the right or left, it is shown that the bottom left and right angles don’t change. Not only the “bigger” triangle doesn’t change, the “smaller” triangle preserves the same angles. The lesson Problem about Angles (1) in the same section introduces a problem to the student, and also provides a hint button to provide additional information. No description is provided but by manipulating the graphics, the student can see how the angles remain the same as one moves the red dot. The creative use of animation reduces significantly the amount of explanation that is required and this can be positive to a lot of students.
For students who are confident in deducing math principles, having the abilities to manipulate graphical representations and to see the effect of these changes instantaneously on the screen will be useful to them. However for students who may require extra help, such tools may seem cold and unfriendly. The hints provided in some lessons are useful, but in cases where the students still do not grasp the principles after exhaustion of the hints, additional help from the teachers or other students should be supplemented. Learning math does require one to process the concepts individually, but there are often cases where collaborative learning helps reinforce the student understanding and comprehension of the subject. The problem with many of the software visualization systems is that the students are left on their own in the learning, and it misses out on the collective engagement and knowledge sharing with peers.
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