I’d like to blog about an important problem solving skill: The ability to make new symbolic representations. Fundamentally, it’s a language skill. Languages are symbolic representations, and where would the human race be without language?
I suppose one could theoretically imagine an individual without language skills inventing a cellular phone, but I think such an event is even less likely to occur than witnessing a chimpanzee produce Hamlet on a typewriter. People who came before us left us a legacy of symbolic representations. This post itself contains 1070 of them, some quite modern like the word “blog” (at least, in the modern sense). Symbolic representations help us organize our ideas and enable us to reach higher levels of sophistication in our thoughts.
In mathematics, the act of creating new symbolic representations happens routinely. Look at any random mathematical post by Professor Terrence Tao, and you’ll frequently find the construct “Let X be…,” where X is some symbol. This symbolic naming skill is important at all levels of mathematics, not just at the professional level where Professor Tao operates. One early, math-specific, instance of this skill is the concept of the variable. Students who haven’t mastered variables typically get stuck solving “word problems.” Here’s a common example:
Heather routinely put her pocket change in a big jar. Because she found quarters useful for parking meters and laundry, the jar contained only pennies, nickels, and dimes. After several years, the jar became filled to the brim and she decided to cash the coins in. When she stacked them, the coins formed a column 3,593.4 mm high. When she cashed them in, she learned that there were 2,340 coins with a total value of $141.26. How many pennies, nickels, and dimes were there? (According to the U.S. Mint, the thicknesses of pennies, nickels, and dimes are 1.55 mm, 1.95 mm, and 1.35 mm, respectively.)
For people who lack facility with variables, the idea of writing “Let P be the number of pennies. Let N be the number of nickels. Let D be the number of dimes,” is beyond their ken. For this particular problem, though it may not be absolutely necessary to introduce variables to solve the problem, it helps enormously. The best way to judge the value of variables for solving this problem is to try to solve the problem yourself without using variables. (If you’re well-versed in this kind of algebra, you’ll probably find it hard to think in a way that isn’t influenced by your facility with variables.) The use of variables to solve this problem helps to isolate the mathematics from obfuscating details and organize your mathematical manipulations.
In fact, students who cannot introduce symbolic representations for the numbers of pennies, nickels, and dimes often find themselves completely stalled. They stare at the problem with no idea how to proceed. (I should state that the ability to introduce the variables is not a panacea. Problem solving requires additional skills.) Yet, these students can succeed in solving more transparent versions of the same problem. For example, though they lack facility with variables, they can usually solve the following:
Heather finds some coins in her pocket. She sees only pennies and nickels. There are 7 coins total and they amount to 15 cents. How many pennies and nickels are there?
For this problem, one can avoid variables and reason like this: “If there were 7 pennies, that would be 7 cents, so that can’t be right. If there were 6 pennies and 1 nickel, that would be 6 plus 1 times 5 or 11 cents, so that can’t be right. If there were 5 pennies and 2 nickels, that would be 5 plus 2 times 5 or 15 cents…and that’s consistent with the given information, so the answer is 5 pennies and 2 nickels.” Perhaps, such students would attempt this kind of procedure with the first problem, but they would surely give up before finding the answer. (If you’re skeptical, try this method on the first problem yourself!)
So far, my point has been that possessing the ability to make new symbolic representations (such as the act of introducing a variable) is extremely important if one wishes to be able to solve more challenging problems and understand more advanced mathematics. Now I’ll tie this in with the MCAS.
Take a look at the 2011 MCAS Grade 10 Mathematics Test. See if you can solve all 42 problems without ever using the skill discussed in this blog. You will find that it’s no problem to do so. For none of the problems is there any great benefit to introducing a new variable or making a new symbolic representation. For example, here’s problem #39:
Ken and Jerome went to the same electronics store.
- Ken bought 2 video games and 1 DVD for a total of $105.
- Jerome bought 1 video game and 4 DVDs for a total of $105.
Each video game cost v dollars and each DVD cost d dollars.
Which system of equations can be used to find the cost, in dollars, of each video game and each DVD at the store?
In this problem, the test question itself introduces variables so the student doesn’t have to.
A person can get a perfect score on the 2011 MCAS Grade 10 Mathematics Test without having the extremely important skill of being able to make a new symbolic representation.
In fact, a person can lack this skill and get good grades in math classes and high scores on math standardized tests throughout K-12. Turn it around and we see that students who present good math grades and test scores may still be lacking at least one extremely useful problem solving tool that they will need if they wish to study more advanced mathematics. I’ll dare say that this skill is critical in all research arenas. (I’m reminded of when Professor Eric Lander introduced the term “chromosome” at about 44:35 of this videotaped lecture on genetics at MIT: “What is the appropriate scientific procedure when you have no clue what something is?… You need to give it a name…”)
I’ve discussed only one critical skill, but problem solving involves many such skills that are similarly ignored by standardized tests. Our society seems obsessed with simple, “scalable” solutions to all our problems. Some day, artificial intelligence may improve to a point where it will provide a scalable solution to quality math education for all, but until then, we must continue to support excellent math mentors. At this time, it is only skilled mentors that can diagnose what standardized tests are blind to and help students equip themselves with these important problem solving tools.