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The Power of Visualizations

 

There is one really important secret to mastering Mathematics: visualization. Math is mostly about shapes, graphs, and counting. One thing in common with all of these is pictures. Pictures are really important, from learning Math concepts to solving problems.

 

 

How do you make sense of sets? You draw Venn diagrams. How do you make sense of functions? You draw graphs. How do you make sense of trigonometric functions? You draw triangles. The list goes on and on. Teaching at the college level, I have noticed that a lot of students struggle with basic Math concepts simply because they do not have the right picture in mind, or any picture at all. These issues could have been avoided if they had used pictures to understand certain concepts at earlier learning stages. When I was in high school, I used to draw pictures to understand certain Math concepts. It helped me develop a very strong foundation in Mathematics. When teaching, I try to use as many pictures as possible and encourage the students to do the same. You don't need to be an artist; a simple sketch would do.

 

 

If you are stuck on a Math problem, try to picture it out and draw it—a simple picture can work miracles. I have seen students struggling with really easy problems, but once they have a picture, everything seems really obvious. Not all types of Math problems need pictures, but for those lengthy word problems, a picture could help save the day. Most students hate those word problems because they seem really confusing and are often hard to relate to the Math they actually know. It would say something like "Bob starts out at city A at this much speed, then he meets Alice traveling in the opposite direction at this much speed...". I would suggest you draw this out on paper, one scenario at a time, see how they relate to each other, and you will feel at home. When I was learning Math I couldn't do without a picture for these types of word problems. So how do you learn to draw these? Once again, practice. If you practice enough, you would learn really well to do quick sketches.

 

 

A really important type of picture are graphs. Remember, Calculus starts out with graphs. Some students struggle to understand certain concepts such as "gradient", "slope", "slope at x = 1", etc. Drawing graphs can be quite messy and, at times, laborious. There are a lot of free online tools that help you draw and play around with graphs. Geogebra and Desmos are two such freely available tools. Use these tools to play with these Math objects, either 2D or 3D. Give it a try. You will learn a lot. Even as a Math PhD student, I still play around with these; sometimes they help me in my work. 

 

 

In the future, I plan on doing some video illustrations to show how pictures and graphs can actually be useful. So stay tuned for more on this.