Have you ever met a mathematician? Have you wondered how they got to be where they are, how incredibly smart they must have to be in order to work with the most complex mathematical concepts? They might say that math is just logical, or that it’s just the way their brain works. But what does that mean, exactly?
We’re going to explore what’s behind that logic—what really goes on inside the mind of a mathematician, and how they think about the math they work with. How can we approach problems like mathematicians ourselves? Is the logic that seems to come so naturally to them accessible to the rest of us, too?
Break It Down
Sometimes, problems come to us like a tangled ball of yarn with pieces wrapped around pieces and tangled into a knot that seems untangle-able. Give that ball of yarn to a mathematician, and he’ll work through the tangles systematically. Mathematicians start with one string and follow it to the next one, and the next one, and the next one, until they find a loose end they can separate from the rest. They will tease apart section by section, one section at a time.
This is analogous to how mathematicians approach problems in their work. Any mathematician will tell you that key to their problem-solving strategies is breaking things down into pieces—and this is one of the most fundamental strategies we teach students facing math queries. Starting from one place, follow the train of variables until you can separate one out. Then, tackle one piece at a time.
Bar Models, Visualizations, and Word Problems
Our favourite way to teach breaking problems down is to use visuals, such as bar models. Bar models are powerful, flexible tools that enable students to see each piece of the puzzle as a separate item, positioned in context to the other parts of the question. We introduce bar models to our youngest students performing the most basic arithmetic so that they are ready to apply those modelling skills when questions become more complex.
Here is an example of a first-level two-step word problem, where bar models help students separate out the multiple steps and what’s required of them:
Annie and Leroy have 628 bouncy balls. Annie has 152 of the bouncy balls. How many more bouncy balls does Leroy have than Annie?
Untrained students, confident in their budding arithmetic skills, rush to calculate without stopping to consider what exactly they are calculating for. Many students do not realize that this is a two-step problem because they do not take the time to break the problem down and absorb what is being asked. Students may write down “628 — 152 = 476” for their answer with no additional explanation given.
We strive to train students to take systematic, visual approaches to questions—even the very simplest ones—so that they get in the habit of modelling and will be much less likely to miss key bits of information and will be able to pinpoint exactly what they need to calculate. Even further, modelling helps to develop a visual intuition so that students become more adept at breaking problems down naturally in their minds.
A bar model approach to the question above would look like this:
Two bars are drawn, one larger and one smaller. These bars represent the two people with bouncy balls in the question. The next step is to fill in the information given in the question and identify specifically on the model where the final answer is that we’re looking for. In Annie’s bar, the smaller one, 152 will be written, and the other bar will be left blank. The 628 will be written to the right of the two bars to indicate it is their combined total. And where is the final answer? It is the difference between the two bars, indicated by a question mark at the gap between the two bars—the part of the big bar that overhang the small bar.
Students quickly deduce that in order to find the gap, one first needs to find the amount for the bigger bar, and the two-step process evolves naturally from there. These students understand why they are subtracting 152 from 628: to find the amount of the smaller bar as a precursor to finding their final answer.
It is not difficult to see how this would be helpful with much more complicated word problems:
425 men, women, and children attended a theme park first thing in the morning. At noon, one quarter of the women take half of the children home. The remaining number of women is now the same as the number of men, and the remaining number of children is now one third the number of remaining women. Find the number of men, women, and children that attended the theme park first thing in the morning.
It is clear than an order to solve this question, one must compute ratios among the numbers of the different groups of people and work backwards in time. This question becomes overwhelming when a student is trying to keep all this information in their brain at once, juggling all the interdependencies while trying to solve for one variable.
Breaking the question down into its independent parts and finding the things that can be independently solved becomes critical. From there, pieces that are dependent on those freshly-solved variables then become possible to tackle. Bar models are one way to empower students to break questions down like this in a visual problem-solving way.
From the Classroom to the University
Bar models are of course not the only approach to being able to visualizing a problem’s component parts. We love them because they are a fantastic way to teach young minds how to intuitively encounter the arithmetic questions they face in early education. However, mathematicians working through pure mathematics problems in their university departments may not be drawing bar models as liberally as we do.
That being said, an important principle is being exposed. Mathematicians are fluent in visual thinking when it comes to math. They can see exactly the variable that they are going for. They can see exactly the facts they need in order to solve for that variable. And they can see exactly which of those facts they have enough information to solve for independently and which will create a domino effect enabling them to solve the rest. Or, they can see where there are holes in the information they have, making the question an inspiration for further exploration.
This ability to tackle problems in their component parts affords mathematicians with a higher comfort level with more complex sets of information and even with ambiguity. This is not only invaluable in math, but it is a valuable decision-making strategy in all kinds of facets of life. Stopping to tackle only what you can, and tuning in to what needs to be solved first, is a recipe for fighting overwhelm in any area. And that seems like a very logical pursuit.