THE POWER OF VISUALIZATION
from the Book: “From Math Anxiety to Math Mastery in 6 Stages: Encoding Mathematical Symbolisms”
A picture is worth a thousand words…and numbers, when we try to think mathematically. One has to use a lot of words and numbers to explain the difference between division as equal sharing and division as equal grouping. It is because mathematics engages both numerical and verbal reasoning. The two run alongside, entwine, criss-cross, meet or part ways. All these behaviors are deliberate and precise, and a slight change in accent can cause a change in meaning. When explaining the difference between “factors” and “factoring” a teacher has to consider the mathematical context which itself may require clarification.
[All that I have said so far can be demonstrated via one or two carefully crafted, evocative visuals.]
Mathematical concepts provoke visualization and a lot of them consist of dynamic imagery, of ideas caught in motion, connections that remain stable and understandable while moving from one state to another. They collapse when the motion ceases, or slow down below a certain threshold of speed. Of course, the visualization has to make sense even when not mathematical. For instance, you can show a static shot of a figure jumping off a 60 feet high diving board into an olympic swimming pool. The next figure is shown falling vertically, head down, arms outstretched. The last one shows the splash that follows the dive, with just the feet showing. If this was shown visually, you wouldn’t have to read so many words to visualize it. You would get it at a glance, within seconds, freeing up time to think about it, and even feel it.
If the last shot showed the diver in a horizontal position with arms fully outstretched, as if flying above the water, it will leave you wondering: “That makes no sense”. It’s because you are reasoning visually.
Visual reasoning holds the key to unraveling math concepts. In long, broad strokes, the images displace pages of printed words while inviting both verbal and numerical reasoning. When combined with principles of Universal Design in Learning (UDL), the learning of mathematics can turn into a pleasantly immersive experience.
For instance, imagine a tube with an open top in frame 1. The transparent one is shown on the left, and the opaque one on the right. Above it is a clutter of jellybeans. The jellybeans are stored in the tubes. The tubes are labeled “T” for Tube:
Myra and Kian are playing an interesting game. Myra has 3 empty tubes and Kian has 2:
Then Myra takes 3 jellybeans, and Kian takes 5:
Goal of the game: Each has to put the same number of jelly beans inside each tube so that the total number of jellybeans (inside and outside the tubes) are the same for Myra and Kian. How many jellybeans should go in each tube?
They began first by putting 1 jellybean in each tube:
It didn’t work, because Myra got 6 and Kian got 7 jellybeans. They were not equal. So they put another jellybean into each tube:
It worked with 2 beans in each tube. Each of them got 9 beans. Is there a smarter, quicker and surer way than trial-and-error, to find out how many beans should go in each tube so each ends up with an equal number of jellybeans?
Here is how:
Rule for Step 1: Remove the same number of tubes from Myra’s and Kian’s side. Make sure it is the most you can remove:
After removing 2 tubes from each side, Myra is left with 1 tube and 3 jellybeans. Kian is left with just 5 jelly beans.
Rule for Step 2 : Remove the same number of jelly beans from each side. Make sure it is the most you can remove so that 1 person is left with just jelly beans:
After removing 3 jellybeans from each side, Myra and Kian are left with the same number of jellybeans, 2 each:
The solution is staring at us. Each tube must have 2 jellybeans.
Let us check to make sure:
Yes, it is correct. Each gets 9 jelly beans. The earlier trial-and-error approach gave the same solution.
Let us look at it mathematically. The problem being posed is shown in the left frame. The right frame (14) shows Step 1 when two tubes (or 2T’s) are removed from each side:
In Step 2 (15), three beans (or 3) are removed from each side:
The answer is T = 2.
In Algebra we show it very formally in this way. Here, x represents a tube, or T:
Now try this one. Find out how many jellybeans should go inside each tube:
Try the trial-n-error method first, then the smarter one, using the algebraic approach of removing the same amount from each side, first the T’s then the beans. Of course, you can get the answer within minutes just by studying the graphic and scribbling numbers on paper. You will be reasoning visually, verbally, numerically and even abstractly.
But mathematics is not just about getting the right answers. It is about declaring one’s intention and planning, about demonstrating one’s skill at making executive decisions that can be defended by logic and reasoning. It is by recording these as mathematical statements (as in frame 16 above). At the end of the day, the easiest way to dispel anxiety is to know what you are doing, why you are doing it, and how you chose to do it. It fosters respect for other innovative ways of doing it.
from the Book: “From Math Anxiety to Math Mastery in 6 Stages: Encoding Mathematical Symbolisms”