Back to: Module: Helping Students Do Math
Slide 1: Visual math refers to the graphs, charts, and pictures through which math facts, relationships, and ideas are displayed. Welcome to this webinar with Daniel Showalter.
Slide 2: Visual math uses graphs, charts, and pictures to display math facts, relationship, and ideas so that we can actually see them, all at once. With these displays you can show students what’s going on with all the numbers. Let’s start by taking a look at a couple of famous examples of visual math.
Slide 3: The chart on this slide was drawn by John Snow in mid-19th century London. Each bar represents a case of cholera during a lethal outbreak. (Cholera is a disease carried by water that has been contaminated by sewage. Before Snow proved what was happening, no one knew how the disease was spread!) Snow collected his data and displayed it visually to show the source of that outbreak as a particular water pump used in the neighborhood. This knowledge has saved millions of lives. But it was the visual display that established the facts (the particular pump), the relationship (those who got sick all used that pump), and the idea (cholera is caused by contaminated drinking water).
Slide 4: Another famous example is the chart on this slide drawn by Florence Nightingale during the Crimean War. As a nurse, Nightingale was appalled at the proportion of soldiers dying from preventable causes such as poor sanitation. In this chart, the blue and gray areas represent the preventable deaths and the pinks represent the deaths in battle. Nightingale’s chart prompted major changes in British Army policy. Such displays are not just practical, they change minds and save lives. It’s math—and pretty simple. The charts created by Snow and Nightingale were nonetheless quite creative and unique. Nobody else had put this stuff together, after all.
Slide 5: Because it would be impossible to cover all the ways that math is represented visually, let’s focus on a few of the most common types. Many of these, like the ones developed by Snow and Nightingale, provide graphic illustrations of data.
Slide 6: It is helpful to know some basics about data in order to understand what these illustrations can show. Data are bits of information, in context, that can be used to understand a situation better. The context part is important. For example, 3, 1979, and 23 are just numbers. Three inches, the year 1979, and 23 deaths due to cholera are numbers in context. Data can be expressed in categorical or numeric ways.
Slide 7: We express data that can’t be counted or measured as categories. Some examples of categories for grouping the properties of objects, people, plants, and animals are car color, gender, type of tree (evergreen or deciduous), and dog breed. Categorizing is a mathematical operation, however. It imposes a kind of order that helps one think.
Slide 8: When something can be sensibly counted or measured the data become numerical. Some examples of things that can be counted (that is, measured in a quantitative way) are speed, height, number of siblings, and salary: 100 miles per hour, 72 inches, 5 siblings, $30,000 per year. So, there are basically two kinds of data: categorical and numerical. Math deals with them both.
Slide 9: We can use variables to describe both categorical and numerical data. In math, a variable is something that is able to vary. For variation, though, you need a collection of data and not just a single case. For instance, in a collection of data about dogs, we might have 15 German Shepherds, 3 Irish Wolfhounds, and 1 Siberian Husky. Those are categories, but what’s in the categories varies across the collection (called a “data set”). In the case of numerical data—such as speed, height, number of siblings, and salary—the values would be numbers. In a collection of numerical data about five people’s height, for instance, we might have 50 inches, 32 inches, 76 inches, 21 inches, and 62 inches. Of course, you might also be able to guess the age of some of these people! Anyway—that’s how data sets and variation work.
Slide 10: One common way of presenting data for categorical variables is a table. Here, you can see examples of a one-variable table and a two-variable table. Tables like these often have a final row or column that provides a total sum of the contents within that row or column.
Slide 11: Another way to present data for categorical variables is a pie chart. As the name suggests, a pie chart looks like a circular pie. The size of each pie slice represents its portion of the whole. A pie chart helps us visualize the amount of data in one category relative to another category. For example, in this pie chart, we can see that roughly 80% of paraprofessionals are female, which is four times the percentage of male paraprofessionals.
Slide 12: Many types of visual math make use of axes. An axis is a reference line on a graph. The plural of axis is axes.
Slide 13: One of the simplest uses of axes in visual math is a bar chart. In a bar chart, the values of a categorical variable are listed out across an axis. Then, the number of cases belonging to each category is shown in the form of a bar. The longer the bar, the more cases fall within that category. Here is a bar chart showing the number of vegetable servings eaten daily by each member of my family. You can tell that Ellie is quite a picky eater!
Slide 14: A line graph also allows us to present data on two numerical variables. We first plot the data points, and then connect them with a line to show a trend. Here is a line graph showing the growing shoe collection of my daughter Ellie. Each dot represents the pairs of shoes she had by a given birthday. By connecting these dots with lines, we can estimate values even when we don’t have data. For example, we could guess that Ellie had about 6 pairs of shoes when she was a year and a half old.
Slide 15: Up until now, we’ve been looking at examples of using visual math to represent data. Visual math can also be used to help simplify more abstract concepts.
Slide 16: Let’s say that, to qualify for a certain government program, you must have an annual income less than $20,000 a year and either have a physical disability or be older than 65. For a given person, each of these three qualifiers could be either true or false. A truth table, as shown in this slide, can help organize all the possible combinations of trues and falses to help us make a decision. Notice that this same truth table could be used for any situation where we need to meet a certain condition and then at least one of two other conditions.
Slide 17: Visual math is particularly useful in geometry to help understand shapes and relationships. Consider the following abstract description: A circle of radius 2 is the set of all points in a plane that are two units away from a given point. Although the definition is precise, it’s much more helpful to draw a circle and label the radius as shown on this slide! The most difficult part about drawing or reading diagrams is translating accurately between the words and the picture.
Slide 18: There are many other examples of visual math not covered in this webinar. You are certainly not expected to know about all of them. There are, however, some specific things you can do as a paraprofessional to help students use visual math. When they are working on an abstract problem, ask them, “Is there a way to show this problem visually?” If they are dealing with a problem that is already in a visual format, ask them to explain in words what is meant by specific parts of the graph or diagram.
Slide 19: You’ve probably heard the saying, “A picture is worth a thousand words.” This is just as true in math as it is in journalism!