Easy as 1,2,3

By Jhariana Floyd

It is no secret that math tends to be the subject that most students struggle with. There are numerous reasons as to why that is so, but there are steps that can be done to develop stronger math skills. Math teachers are taught to present concepts in CPA format. That means concrete examples, then pictorial examples, and finally abstract examples. It is believed that this allows students to best comprehend new material. However, there are many debates on how information should be delivered to students. It is believed that it is best to teach information with concrete facts and work up to abstract material. Think back to when you first took algebra. Sorry if I am forcing you to relive horrific memories., but it’s to help you.  You were prepared to learn algebra your whole life through the use of concrete methods. Your teachers taught you how to solve equations, then how to graph equations, and then how to guess what a line would look like given facts. Guessing the line requires mental imagery, and I’m going to help understand how mental imagery contributes to one’s success in math.

Breaking down math visuals to gain a better understanding.

Research has shown that it is best to teach both, for students tend to use all algorithms for easy problems, but they drew images for problems that were harder to solve (1). Think back to fractions as a child. When the problems were easy to solve, then you would most likely solve the problem by showing your work mathematically. However, when the fractions got harder, you would probably draw out pictures to assist you. At least that is what most of my students do, and that is what I did as a child. Who knows, you could be a rare case. Additionally, “research has proven that graphics are effective learning tools only when they allow readers to interpret and integrate information with minimum cognitive processing” (2). This leads me to believe that students store facts better than visuals. It makes sense. When students are first learning how to count, they use their fingers because it is easier to add with visuals. But, by the time students learn how to multiply, they are able to look at numbers and answer the product of the numbers correctly. Learning both is beneficial because each serves great importance for different scenarios.

What I have learned from my educational experience at Eureka and my teaching experience at other schools is that the educational system is requiring teachers to teach students how to practice solving problems in multiple ways. One way is through algorithms, and the other is through the use of drawings. Back in my day, I never was required to use drawings. I could use them, but I had to show my work using algorithms. Is there a right and wrong way to teach math? Many can agree that everyone should not teach math or any subject for that matter, but is there a right and wrong way to deliver math concepts? This hits home for me, for that is my future career. And if you plan on helping your future or present children with math, it should hit home for you too.

My advice to you for math is to write notes as simplistic as possible to remember the facts. I say that because we are able to remember facts easily. No one remembers notes that are too detailed. When students copy notes word for word, they are thinking about the process. However, it is important that you also relate them to pictures or symbols. Research states (3). So next time you’re graphing or working out a problem, write down the steps you take. Also, write down the importance of the steps. It will also help with studying, for it will give you a deeper understanding of the content. In the past, when students asked the importance of the material they were learning, teachers would often say, “Because I said so.” Some would flat out say, “I do not know.” Now teachers have to learn the meeting of all the content they teach. For example, do you know why you have to flip the fraction when diving fractions? Probably not. Teachers know how to teach the reasoning behind things like that. I’m not saying your school teachers failed you, for you clearly made it to college. However, they did not prepare you well. They taught you how to spit out facts without reasoning. To be fair, it is easy for students to create mental images from facts. But adding an already created image is beneficial. Think about watching a movie and reading a book.

To clarify, I’m no mathematician, but I am going to help you pass math in college. Hopefully. I’m no miracle worker either. It’s going to take time outside of the classroom as well. I was once told, that for every credit hour a class is, you should study at least 3 times that outside of class. Let’s say you are taking a four-credit-hour class, you technically should be studying twelve hours outside of class for that one class. If I am being honest, I have never done that. I spent many hours getting help from my math teachers outside of class but never 12 hours a week. Personally, math has always come easy for me. That’s why I’m making a career out of it. I always listened to the lecture, watched the teacher work out problems, and then write notes. This process made me retain the information gained. This method helped me become successful in all my classes. Research has stated “I have found that it’s a combination of both that works best. Everyone has their own interpretation of pictures but it’s the propositional aspect that makes it everything come together” (4). When we visualize graphs or symbols, it does not mean much if there is no meaning behind them. Including annotations for visuals will help remember those images. Additionally, the spatial representation is essential for reasoning is a crucial part of mathematics (5). Word problems are best solved when one can visualize a solution. Additionally, some problems require the visualization of the information provided to solve the problem. For example, if a problem asks to find the area of a slice of pizza, you would need to visualize a pizza to realize that pizzas slices are usually triangular. When studying for math, you can improve your mental imagery by developing an understanding of the images. It is not beneficial to just remember steps or pictures or symbols. The more understanding of a topic will help to create stronger mental images. “Many highly original and significant creations of the human mind have been largely the result of nonverbal mental representations” (6). In conclusion, no student is bad at math. They all have the capabilities to excel in mathematics. Many are just studying and gather information in unsuccessful ways.  By focusing on creating spatial and perceptional mental images, one will become an expert in math in no time. Probably not an expert, but a better student.

Rewriting notes to be understood better


  1. Lowrie, T., & Kay, R. (2001). Relationship between Visual and Nonvisual Solution Methods and Difficulty in Elementary Mathematics. The Journal of Educational Research,94(4), 248-255. Retrieved from http://www.jstor.org/stable/27542329
  2. Vekiri, I. (2002). What Is the Value of Graphical Displays in Learning? Educational Psychology Review,14(3), 261-312. Retrieved from http://www.jstor.org/stable/23363549
  3. Clements, K. (1982). Visual Imagery and School Mathematics. For the Learning of Mathematics, 2(3), 33-39. Retrieved from http://www.jstor.org/stable/40247750
  4. Psych 256: Cognitive Psychology FA14. (n.d.). Retrieved from https://sites.psu.edu/psych256fa14/2014/12/12/spatial-vs-propositional/
  5. Pearson, J., & Kosslyn, S. M. (2015, August 18). The heterogeneity of mental representation: Ending the imagery debate. Retrieved from http://www.pnas.org/content/112/33/10089
  6. Clements, K. (1982, March 3). Visual Imagery and School Mathematics. Retrieved from https://pdfs.semanticscholar.org/563b/379a2b9ea0b1d4fad0130c4ac2833a6a4308.pdf


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