We started exploring last month how we might approach problem solving in a way that’s worthy of a mathematician. Mathematicians tend to be highly logical people, unfazed by complexities and details and high-level concepts that leave the rest of us shaking in our thinking caps. We saw last month how we might take their approach in breaking problems down to tease out what might otherwise be overwhelming complexity in a problem. We saw how identifying dependent relationships within a problem can help us prioritize and isolate exactly what we have the power to solve, what can be solved next, or where we might need more information.
This month, we’re going to explore another key strategy that a mathematically-minded individual might take to problem solving: looking outside the question itself and seeing how it compares to more familiar situations. Mathematicians are masters of making analogies.
Metaphors, analogies, similes, allegories…we are no strangers to making abstract connections between scenarios. We use them to turn unfamiliar situations into something more approachable, and our kids do it, too. When kids play pretend, they are taking concepts of social interaction, adult responsibility, and how the world works and applying them to a simplified version of the world they can understand and explore. Family dynamics are distilled down to interactions between toys. Morality is explored in the context of a story. Losses and wins in life are acted out through imaginary characters they put on with playmates.
As educators, we are no strangers to using analogous situations to illustrate educational concepts, especially in math. Fractions almost always start out as pies or pizzas. And you’d be hard pressed to find an introductory addition or subtraction lesson working with just plain numbers, instead of imaginary apples or balls or crayons.
The idea behind an analogy is that it takes an abstract concept and grounds it in tangible terms. Analogies can also help justify what is going into a series of calculation steps. Putting a math problem in the context of a tangible situation can make it clear why each algebraic step is necessary, rather than asking students to memorize a series of rules.
Let’s take a look at an example:
Find 1/4 of 5/6.
Now, we can explain to students that to algebraically solve this question, they must multiply the two fractions together. This becomes much more intuitive, however, when we frame it with an analogy:
You’re trying to find 1/4 of 5/6. Imagine this: your mom brought home some leftovers from a potluck, including 5/6 of a cake that was left. Tonight, she announces that your 4 family members get to have it for dessert. Remember, there’s only 5/6 left. We have to split that amount into 4 equal pieces so that you can each have an equal share. How can we do this?
From there, you can explain that to find a quarter of something, you are dividing by four. You can introduce the concept of taking the reciprocal and multiplying, which gives you a basis for the algebra of multiplying by a quarter.
Now all of the sudden, there is rational justification for the math.
Mathematicians do this sort of thing instantaneously, or even subconsciously. To them, multiplying the two fractions together in this case is perfectly logical because of the implied underlying scenario of finding a fraction of a fraction. Analogies allow calculations to take place not just on numbers, but on three-dimensional entities governed by relationships.
The better we get at recognizing mathematical questions as familiar situations, the more we will see the same scenarios repeated over and over again. As suggested above, once a person gets really good at recognizing what a fraction of a fraction means, they don’t necessarily need to walk through an allegory every time they’re approached with a new question.
The human mind is adapted perfectly to picking out patterns. It’s how we learned to survive, recognizing vulnerabilities of prey or threats of predators. It’s how we develop expectations in social situations. It’s even how we’re able to communicate with complex written or spoken language.
And pattern recognition allows us to solve mathematical queries and conceptualize deep mathematical truths that govern our universe. No other species (as far as we’re aware) contemplates the commanding forces of the cosmos.
Pattern recognition is an essential skill that we love to practice at Mentorhood. Picking out patterns allows students not only to solve problems faster, but to make connections with deeper principles. Take the multiplication tables, for instance. The better students get at recognizing the repetition and patterns in the times tables, the faster they’re able to solve questions. Moreover, they’re able to start understanding how compounding relationships work. This will be essential when they first try to understand exponents or logarithms.
Mathematics are riddled with patterns, and mathematicians have the ability to take what they see working in one scenario and apply those mechanics to a similar situation. They are masterful connection-makers. On the one hand, this helps expedite the calculation process. And on the other hand, this allows for the drawing of new conclusions and parallels.
Getting used to framing problems in familiar terms helps to solidify the logic behind them. And recognizing similarities between problems allows new inferences to be drawn. Both of these strategies make connections between abstract concepts and familiar situations. This is a powerful tool for revealing underlying principles and can be useful for solving new types of problems that have never been seen before.
There is something comforting in the familiar—we are wired to gravitate towards it. Familiarity also gives rise to confidence, which is something we delight to see in our students. The more we can encourage students to see concepts they have already mastered in the math problems they are faced with, the further they are inspired to believe that they are up to the challenge. With a bit of practice, students can become master connection-makers, too.