Let’s face it, word problems in mathematics are popularly seen as, well, terrible. Students typically recoil in horror from any problem that is actually communicated with… heavens no!... sentences. Why is this?
We think that one of the reasons people hate word problems is that they haven’t seen many good ones! In particular, word problems have a habit of suffering from bad context. The context of a problem is the story that is being told through the problem: the setting of the problem. And just like novels, word problems are better when we can find something to relate to in that setting. Finding things in the problem that we can relate to puts us on the solid ground we need to untangle that story and begin solving!
For example, consider the following speed problem:
A boat in a river travels 10 miles with the current in the same amount of time it travels 7 miles against the current. If the speed of the current is 3 mph, what is the speed of the boat in still water?
We may not have ridden in a boat, but we all know what they are, and we can certainly imagine a boat traveling with, or against, the current of a river. And if we really think about it, it makes sense that a boat will travel faster when it goes in the same direction as the current, as compared to moving more slowly when it travels against it. Kids often experience these ideas through soap bubbles in the wind - it’s much easier to blow bubbles with the wind than against it! Because the setting is relatable in a very direct way, it gives us ideas on how to begin the problem.
So, a believable setting can provide something relatable to latch onto and help ground our thinking, but not everything that’s relatable is necessarily believable! In other words, all kinds of contexts can be relatable, whether they be real-life or fantasy. As an example of something decidedly less than real-life, here’s the beginning of this week’s Problem:
A hungry kangaroo lives in the forest, and needs to go forage for food. There is a straight east-west road that passes right in front of his house. The kangaroo only travels on that road. To move, the kangaroo hops, and his journey is just a sequence of hops, one after the other. Whenever he’s ready for another hop, he can travel west, or east, but only along that road.
Kangaroos don’t live in houses, and don’t travel along roads to find their food, so the context isn’t particularly believable, but it is relatable in a number of ways. For one, you can imagine yourself moving around, as the kangaroo. And just imagining that movement gives you an idea of what mathematics might be involved in the solution to the problem (and we don’t even know what the problem is yet!). A straight road reminds us of a straight line, and movement on a straight line reminds us of certain arithmetic operations, which leads to other mathematical ideas that will be important when we finally begin to tackle a solution.
So what about problems with bad context? Can you have a real-life setting for a word problem that gives totally unrelatable context? All too often, that is actually the case! Here’s a great example of a trigonometry problem, that supposedly gives an “application” of trig in real life:
A window washer is washing windows on the side of a skyscraper, admiring the view of the other buildings in the vicinity. He looks at a building 340 feet away, and sees that the angle of elevation to the top of that building is 23.7 degrees, while the angle of depression is 45.3 degrees. How tall is the building that he’s looking at?
In some sense this is a real-life setting, as window washers do in fact wash windows, and they do look at other buildings, and they might actually wonder how tall those other buildings are. But we are asked to ingest a number of things that are, honestly, completely unrelatable. Not only is the window washer dying to know the height of that building right now, he is going to calculate the height by using math and his eyes. Just by looking straight ahead, and then at the top of the building, he is supposedly calculating the angle his eyes make when they move. Not only that, he is calculating that angle to the nearest tenth degree. I don’t know about you, but when I see nonsense like this masquerading as something that happens in real life, I’m ready to close that textbook immediately. The context of the problem is absurd, especially when this is supposed to be an application of trigonometry! This problem is not relatable or applied, it is simply silly and contrived.
Bad context doesn’t inspire students to work on problems. Rather, it inspires students to see word problems, and applications, as silly, and confusing, and a waste of time. We can do better.
As we live our lives, we encounter many problems at work, at home, while shopping, and everywhere else. When we meet these challenges, because they are actually part of our lives, they are believable and relatable. And these challenges are really word problems in disguise! Because every challenge we face in life has a setting, a context, a backstory, that we could write down if we wanted to. When we face down a challenge that pops up in our daily lives, we are solving a word problem! Life is why we need to embrace word problems: life is word problems.
The more our kids are comfortable with word problems, the better off they’ll be in their adult lives. We as educators, and as parents, need to ensure that the mere presence of words in a problem doesn’t make our kids freak out! And to do that, we need to make sure, as often as we can, that the contexts of the problems we’re giving our kids are relatable to their lives in some way. The better we are at word problems, working to understand them and filling out the rest of the story with the mathematics needed to solve them, the better problem solvers we, and our kids, will be!
Ready to try a word problem worth doing? Try your first exSTEMsions problem right now!
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