The Deep Structure of Word Problems | Nurturing Wisdom Tutoring

Jumping into the Deep End: Moving from Surface Structure to Deep Structure

In previous posts, Sara introduced the concept of surface structure & deep structure when it comes to problem solving. She then went on to describe the ways the surface structure of problems can help students understand certain aspects of those problems or concepts. Now Sara goes into more detail about how, through knowledge transfer, we can help students move beyond the surface structure of problems to understand their underlying deep structure.

Transfer (applying past knowledge to new situations or problems)is an essential part of how we define learning. We can’t say that we’ve really mastered a concept or skill until we can apply our knowledge in different situations. The question is: how can we promote transfer?

As I explained in my last post, students have an easier time recognizing deep structure if the surface structure is familiar. The key point here is that familiar surface structure is not necessary to solve a problem – but it is helpful. I’ve noticed that when working with ACT students over the years, most of them have an easier time tackling percentage word problems when they have a familiar context. Take these examples:

1) Sheila had lunch at a restaurant. Her bill was $20, and she wants to leave a 20% tip. How much is her total?

2) A business purchases its inventory at lower wholesale prices and then sells items to the public at higher retail prices. Sheila, the owner of a shoe store, purchases shoes at $20 per pair, and she sells them for 20% more than she paid. What is the retail price of the shoes?

Even though the calculations are identical, most high school students have an easier time with the first problem because they’ve had experience with leaving a tip at a restaurant. It’s rare that you’ll find a high school student who has experience determining retail prices based on wholesale cost.

In the same vein, if my son had had more experience with track and field, he probably wouldn’t have balked at the ‘laps around the track’ addition problem. He’d probably have more confidence tackling this problem: “If an artist uses 2 cans of paint to make one mural, how many cans of paint would he need to make 2 murals?” than he would this problem: “If there are 2 hydrogen atoms in one molecule of H2O, how many hydrogen atoms are in 2 molecules of H2O?” What we can take away from this is that the more exposure a student has to different subjects, the easier it will be to transfer knowledge – and the easier it will be to learn new information, period! Encourage your children to expand their base of knowledge by going to museums, reading books, participating in extracurricular activities, and engaging them in conversation on different topics. This is a case where more is definitely better!

As helpful as a wide base of knowledge is, a student doesn’t need to be familiar with the surface structure of a problem in order to recognize its deep structure. How can we promote knowledge transfer to unfamiliar contexts?

The answer: lots of practice!

This may seem oversimplified, but the truth is that it’s easier said than done. I think most of us know that practice is a necessary part of learning, and that the more practice a student has with different types of problems, the more apparent the deep structure of those problems becomes. After all, when I think of the variety of addition problems that I’ve seen in my lifetime, the amount of exposure my 7-year-old has had seems miniscule. With more practice, the ‘laps around the track’ problem will become as simple to him as it was to me, and the deep structure of addition problems will eventually be unmistakable.

Practice has other benefits in addition to promoting transfer. It helps students retain information for longer periods of time. It also helps basic skills become automatic, which frees up “thinking space” in the brain for performing higher-level problem solving.

The tricky part is keeping students engaged during all that practice! Here are some tips from Willingham in Why Don’t Students Like School?:

  • Break up practice into small chunks of time over a long period. There’s no need to do two hours of practice in one sitting. Instead, try fifteen or twenty minutes a day for a week or two.
  • Vary the problems. Different surface structures not only help with transfer, they can help a student stay interested, too.
  • Mix up the skills in the practice. If a student seems to have a pretty good handle on multiplication word problems but needs more practice, try throwing an occasional addition or subtraction problem into the mix. It’ll keep her actively thinking about the problems.
  • Practice “old” skills in the context of new ones. If students need continued practice with reading aloud, it isn’t necessary to just have them read story after story. Reading aloud could be involved in a lesson about plot or character, or it could be a natural part of acting out a dramatic work.

At Nurturing Wisdom, our guideline is to provide an immense amount of practice, distributed over time, and across varied contexts. Looping, a technique we use when tutoring math, is one way we’ve managed to apply these guidelines with excellent results. I’ve begun using looping with my son for addition, subtraction, and now multiplication problems, whenever teachable moments arise. As a rising third grader, he’s hardly seen the end of stumping math problems – and I’m sure I’ve hardly scratched the surface of moments that I will be surprised and confused as a parent! – but we’re at least one step further along the path of his growth as a student, and I have a much better understanding of how to guide him along in the future.