Visualization, even just within (math) education is quite a broad term. It can be straightforward, for example when creating an instantiation of a math problem. But visualization can also be complex, for example when creating a mind map to aid your studying, or when using an argument map to help make sense of a long text or intricate argument. It might be difficult to immediately see how to implement visualization in your class. I will give you a short overview of the benefits combined with some practical examples, to help you get started!

**Technology in the educational sector**

But let us start at the beginning. Amongst many others, Davies (Concept mapping, mind mapping and argument mapping: what are the differences and do they matter?; 2011) writes that there is not much change in the learning methods of the educational sector. Teaching methods still mostly consist of students reading textbooks and listening to lectures by educators. Even though students are the societal group most involved in technological changes, this trend does not seem to follow into the educational sector. Such a waste, right? Seeing as technology can help keep students motivated, as well as help students acquire the necessary skills for the future. Because let’s be honest, nowadays most jobs involve working on a computer. So why does the educational sector not change to accommodate that societal group?

Well, the truth is they do. But very, very slowly. Technology is creeping into education. Students use laptops to take notes, and teachers use PowerPoints and videos to accommodate their lectures. But we shouldn’t stop there. Technology is a very useful tool, and we need to exploit that tool to the fullest. So how do we exploit technology to our benefit as teachers?

**Visualization in Math**

Jeremiah Ruesch wrote a blog called *“The Power of Visualization in Math”* on Edutopia. He writes about a math problem that was so complicatedly formulated, that his students immediately gave up. He decided to draw images to accompany the problem, thus helping his students to visualize the problem. With help of the visualizations, the students immediately understood the problem, and were able to solve it quite easily. (If you want to read more on the process of visualizing your (math) problems, read the blog post, it offers several very useful tips!)

Visualization can be very helpful to explain math. Since math can become quite abstract, it can be hard for students to understand the issue. When students learn about for example simplifying fractions, they become easily overwhelmed. However, if we were to show a pie chart, and show how and are the same, the entire concept becomes much clearer. Next to the theory being explained with text, visual aids should be implemented, where possible, to help the students. The following image is a great example of how visuals can aid the students learning process.

These visual aids can even be interactive, to offer the students the opportunity to actively partake in the visualization process. Interactive visuals would entail visuals that can be manipulated by the students. This helps students create a deeper understanding of the topic, as well as allowing *deep learning (I wrote about this in a previous blog)*.

**Argument mapping**

Visual aids such as these are very helpful to present abstract concepts. But what if the concept is not necessarily abstract, but rather complex? A great visual aid to help explain complex concepts to students is argument mapping. In short, argument mapping is creating a visual representation of all layers of an argument. It can look something like this:

Argument mapping can give the pros and cons of a certain statement in a visual overview, but it can also help showing the inferential relations between concepts, arguments or ideas. For example, it can show a timeline in a history class, or it could show how to solve a mathematical problem step by step. Argument mapping is not necessarily digital, you can create argument maps with pen and paper. But the positives of digital argument mapping are overwhelming compared to the ‘offline’ alternative. It is easier and quicker to create, and it is possible to add hyperlinks to relevant information in the theory or even in your own notes or PowerPoints of the teacher. In their research on CAAM (Computer-Aided Argument Mapping), Dwyer, Hogan and Stewart (An evaluation of argument mapping as a method of enhancing critical thinking performance in e-learning environments; 2012) claim that there are three benefits to using CAAM:

- It helps students to process information through presenting arguments through two modalities: visual and verbal
- It aids students’ critical thinking process through the Gestalt grouping principle, which helps processing information in the working memory and in the long-term memory
- It aids the students in memorizing and storing information in the long-term memory through presenting information in a hierarchical order

Argument mapping can be done passively, where teachers can make the map in advance, and show that to their students. Through passive argument mapping, students can be shown the entire argumentative process in order to enrich their understanding of a concept. Furthermore, passive argument mapping helps lower the cognitive workload of the students. An argument map is easier to understand and remember than an entire passage of text.

But argument mapping can also be done actively, where students create their own argument maps, as a way of taking notes. The advantage of taking notes like this, is that it forces the student to actively engage with their study matter, to think about the connections and link it to previous knowledge. As a result, students will both remember and understand the subject matter much better.

Argument mapping can be used for every course, whether it is for History, Philosophy, English or Mathematics. So let’s take a more in depth look at how we can apply argument mapping!

## Argument Mapping in Mathematics

So how can we use argument mapping in Mathematics? Well, for one, it can be used to guide students through a complex math problem.

As mentioned before, a type of argument map can be an illustration of the step by step process of solving a math problem. When a student reads the theory, it can be quite complex. So in order to help the student grasp the concept, we can offer them a visualization of a manifestation of the concept. So, we give the student an example of a problem that illustrates the concept. This example is immediately worked out, as to present the students with a visualization of the theory. This worked out example is a form of passive argument mapping. This example can be explained using a literal map, showing the steps with lines and figures. But it can also be done via text. Such an example can look something like this:

Next to this passive form of argument mapping, an active form of argument mapping can also be achieved in Mathematics. For example, SOWISO offers exercises where not only the answer, but the step by step solving process is required. This forces students to actively engage with their subject matter, and helps them to engage in deep learning rather than surface learning.

So in short, when you want to add visualization to your teaching methods, keep in mind the following:

- Explaining an abstract concept? Visualize the concept, preferably with an interactive visual to let students actively engage with the subject matter.
- Think of a real instantiation of the concept
- Visualize that instantiation
- Let students actively interact with the visualization

- Explaining a complex concept? Visualize the concept using an argument map, help students grasp the concept easily, and later on push the students to create their own argument maps. When creating an argument map, make sure to focus on the inferential relationships!
- What is the proposition/problem?
- What are the arguments/following steps that need to be taken?
- How are they connected/what are the inferential relationships?

I want to thank you for your time, and I would love to hear any comments or questions that you might have!

Talk to you soon,

Thijs

Good idea! Have you delved into learning basic math facts by using visual graphs and relationships for easy memorization?

Hi Evelyn, good to hear you like the idea! Sowiso offers these visualizations with a lot of their theory, ranging from basic math facts to complex ideas! PassYourMath does the same.

And the research I read on this subject was not necessarily focused on math education, but more on education in general. However, all articles did show the same tendencies, where visualization greatly improved memory retainment of content and relational aspects.

Hope this answered your question!