The Power of Visualization in Math
“On average, the sun’s energy density reaching Earth’s upper atmosphere is 1,350 watts per square meter,” I explained to my fifth-grade summer school pupils when they were presented with this word problem: Suppose the incident monochromatic light has a wavelength of 800 nanometers (each photon at this wavelength has an energy of 2.48 x 10-19 joules), and the incident light is white. Approximately how many photons strike the Earth’s upper atmosphere in a second?”
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Jeremiah Ruesch created a cartoon depiction of a photon to illustrate his book.
Students were able to better understand the author’s question because of his cartoon photon.
Neither the terminology nor the varying sizes of the different numbers, nor any of the scientific principles addressed in the question, we’re able to go past my students. I had essentially turned off their learning, and I needed a new way to reawaken their interest in the subject matter again. After that, I started drawing on the whiteboard and came up with something a little whimsical, a cartoon photon who inquired as to how much energy a photon possesses.
Students immediately began calling out “2.48 x 10-19 joules,” and they were able to cite the literature in which they had obtained the information. So, the next thing I created was a series of boxes featuring our friend the photon, which I knew was on to something.
What amount of energy would be represented in the drawing if all of the photons in the image below were to collide in a single second?
Drawing by the author, Jeremiah Ruesch, depicts a succession of photons impacting the Earth’s atmosphere in a cartoon format.
When a series of photons strike the Earth’s atmosphere, the equation becomes obvious since the total energy of the photons is known, as is the energy of each photon.
Students immediately recognized that we were simply adding up all of the individual energy from each photon, and then realized that we were performing multiplication. And then they realized that the issue we were attempting to solve was simply finding out the number of photons and that because we knew the total energy in one second, we could compute the number of photons by dividing the total energy by the number of photons.
The idea is that we reached a stage where my students were able to digest the information they had learned. Visual representations were quite effective for these kids, and being able to sequence through the problem while employing visual cues fundamentally transformed their interactions with the task.
For those of you who are thinking along the same lines as I am, you may be wondering, “So the visual representations worked with one problem, but what about other types of problems?” There may be a visual model to solve every problem!”
My fifth graders’ excitement as they were able to not only understand but also explain the problem to others convinced me that it was worth the effort to pursue visualization in math and attempt to answer the following questions: Is there a process to unlock visualizations in math? Does visualization in math require a specific set of skills? And, if so, are there any resources already accessible to assist in visualizing mathematics?
The author, Jeremiah Ruesch, has supplied a chart of math resources for your consideration.
I realized that the first step in unlocking visualization as a scaffold for students was to shift the type of inquiry I was asking myself in the beginning. Starting with the question, “How may I visually express this learning aim?” is a good place to start. This repositioning brings us a whole new realm of possible representations that we might not have considered otherwise. The first step in generating a good visual representation for pupils is to consider a large number of different possibilities.
The Progressions, which were produced in conjunction with the Common Core State Standards for mathematics, are one source for locating specific visual models based on grade level and mathematics standards. A tape diagram is a form of visual model that is composed of rectangles that represent the pieces of a ratio. In my fifth-grade example, what I created was a sequenced procedure to generate a tape diagram. Unbeknownst to me, finding a visual representation of the problem was the key to unlocking my thinking and unlocking my potential. Using a very simple set of questions, you can lead yourself down a variety of learning paths, which prepares you for the next phase in the sequence, which is locating the appropriate resources to complete your visualization journey, which comes next.
When you ask the visualization question, you are preparing your brain to find the most appropriate tool for the targeted learning aim as well as your students’ needs. In other words, you’ll be able to tell more easily whether you’ve chosen the most appropriate tool for the job for your kids. This procedure may be made even easier by using the numerous resources available. I’ve prepared a matrix of clickable tools, articles, and resources to assist you in this endeavor.
In my Visualizing Math graphic, I explain the process of visualizing your math lesson; below that is a collection of visualization tactics and tools that you may utilize in your classroom the next day.
Our role as educators is to create an environment that encourages our students to learn as much as they possibly can, and teaching mathematics in this visual manner gives a tremendous tool for us to do our jobs effectively and efficiently. At first, the act of visualizing mathematics will put a strain on your abilities, but you’ll soon discover that it helps both you and your students learn.