Show a child some tricks and he will survive this week’s math lesson. Teach a child to think critically and his mind will thrive for a lifetime.
Math word problems confound many students for a variety of reasons. Too often, well meaning parents and teachers attempt to help students struggling with these kinds of problems by offering them tricks or shortcuts for getting to the solution without necessarily understanding what’s going on. Just do an internet search for something like “how to solve word problems,” and you will get a slew of Web pages with tips, tricks and strategies like the following:
• Search for key words that will tell you which mathematical operation you should use (e.g., “lost” means subtraction).
• Cross out non-essential information so that you are not distracted by it.
• Draw a picture to illustrate each step of a problem.
Let’s consider these three examples for a moment. To suggest that looking for key words within the text of a math problem is somehow a “strategy” to solving it more easily seems a bit absurd. Basic reading comprehension skills are obviously essential to solving any problem, math or otherwise, where the information is communicated via natural language. No key word mapping is going to provide a fool-proof way of understanding what a word problem is asking. If you’re starting with a word problem that is well designed and written with clarity, there simply is no “trick” to understanding the words.
The other two examples sound reasonable on the surface, but a closer examination reveals problems with them too. To determine which pieces of information are relevant to the solution (as in Example #2), you need to be able to evaluate and analyze the given information. To correctly translate the words of a problem into a step-by-step illustration (as in Example #3), you need to be able to understand and prioritize the information. In other words, tricks like these seem to require the very skills needed to solve the problem without the tricks! Furthermore, a “tip” cannot tell you whether you have enough information to solve the problem, or how to find any missing information. Only critical thinking and logical reasoning skills can help you there.
This approach of relying on “clever strategies” to replace critical thinking runs counter to the IMACS philosophy of teaching children the fundamental skills they need to solve problems throughout their lives, whether they be elementary word problems or complex Calculus problems, whether it’s making correct change or correcting the flight path of an exoplanet-bound space craft.
Meaningful Strategies That Matter
While silly tricks may work for some students as a short-term fix to “just get through the problem,” there are practical and helpful techniques that should be part of any meaningful approach to understanding and solving word problems.
Pay attention to units of measurement. Word problems commonly feature units of measurement. When finding a solution, it’s important to pay close attention to how you are adding, subtracting, multiplying, or dividing amounts in the problem so that you don’t end up combining “apples and oranges.” If you carefully track the units of measurement through each step of your computation, you can compare the units of your answer to what it should be based on the question. This is a good technique to check the reasonableness of your answer rather than one that “magically” solves the problem for you. Here’s an example:
No one would think that the following approach is right
because adding children to cookies makes no sense! The answer requires a relationship in the form of “X cookies per child” – or, more algebraically,
With this in mind, what makes sense is the following:
While we’re on the topic of cookies, here’s another example:
Most people know to do something with ( 4,500 / 3 ) and 5, but what? Looking at the question in the problem, we know that our answer should be in the form of “X cookies.”
What if we set up the equation correctly
but then worked out our answer as follows?
Hmm … 300 cookies per minute per minute? The fact that our answer is not measured purely in cookies as expected tells us that we made a mistake in our calculation.
Use reverse word problems to practice abstract thinking. The process of solving word problems often requires us to connect a real-life situation described in language we know with the abstract version written with variables and equations. As with learning to speak a foreign language, thinking abstractly becomes more natural with practice and regular use, and the earlier you start, the easier it is.
When kids are younger, parents can help them to develop an intuition for abstract reasoning by doing “reverse” word problems in the guise of creative play. Start by writing down an equation such as 7 + ___ = 10. Next, ask your child to imagine a story that goes with this equation. He might say something like, “It takes 10 gold stars to get extra recess, and I already have 7. How many more stars do I need?” For 3 + (8 × 2) = 19, she might say, “When I started robotics, I took 3 introductory classes. Then, I took 2 classes per week for 8 weeks. So, I have taken a total of 19 classes.”
Be sure to keep the level of difficulty appropriate for your child’s math ability. Frustrating him with a computationally hard problem will only defeat the purpose of teaching him to think abstractly.
Word problems can be used effectively to teach important mathematical concepts and to help give real-world context and purpose to what may seem like useless hieroglyphics to some people. But without a firm foundation in critical thinking and logical reasoning skills, a student is sure to be at a disadvantage when trying to evaluate, analyze, prioritize, and synthesize the numerous pieces of information embedded in the narrative. Tips, tricks, and strategies may help at the margins, but there is simply no substitute for genuine understanding.
For clarity’s sake, assume that the cookies that Sally and Suzy bring to class are all identical. Thanks, Henry S.
I totally, 100% agree that silly tricks are the wrong approach to teaching math, yet my impression is that this exact approach is far too common in classrooms.
How can teachers who “get” this message spread it to others not so confident to teach critical thinking and logical reasoning?
One problem here I think comes up because of the pressures of external testing. We need to meet someone else’s firm timeline of what is appropriate with our students, so we use tricks to game the system, and get the accountability people off of our backs. Unfortunately, as you have pointed out, this is in detriment to the actual skills we want our students to have.
The solution is to ignore the external accountability measures and focus on what is best for your students, and then suffer the consequences. If every educator did this, pretty soon you’d see the accountability systems get changed.
Thanks- great post. The issue extends well beyond solving word problems. We ignore understanding and teach meaningless procedures often in mathematics eg just add a “0” when you multiply by 10 or when dividing fractions just flip the 2nd fraction and then multiply.
I always insist on building a rock hard foundation when it comes to learning mathematics; a student can become highly efficient in problem solving only with sufficient practice and exposure. There is simply no shortcut. Period. Peace.
This and many great math posts can be found at the ‘Math Teachers at Play’ blog carnival #47, hosted by Math Hombre. Here’s the link >>http://mathhombre.blogspot.com/2012/02/math-teachers-at-play-47.html.
Check out another great math blog carnival, Mathematics and Multimedia Carnival 20, at http://geogebracentral.blogspot.com/2012/02/mathematics-and-mutlimedia-carnival-20.html.