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.

A Closer Look at Surface Structure

Applying past knowledge to new situations or problems is called transfer.

As many of us teachers and parents have experienced, getting a child to transfer knowledge isn’t always easy. The ‘laps around the track’ problem that I gave my 7-year-old son was a poignant example for me. Students will often focus so much on the unfamiliar surface structure of a problem that they miss the deep structure, and they don’t realize that they actually have the skills to solve the problem.

Why do students put so much emphasis on surface structure? Because it’s the clearest part of the problem. The surface structure is written out and obvious, whereas the deep structure is hidden and unclear. It seems more efficient to work with obvious and concrete information rather than take haphazard stabs at what the deep structure might be.

It is important to remember, however, that paying attention to surface structure – the content of the words, the meaning of the story – is actually a necessary step when we’re learning new information. We learn new information by connecting it to information we already know.

For example, the other day my son was reading a children’s non-fiction book about murals. I know that he already has some background knowledge on the concept of art: painting and sculpture are kinds of art, Picasso and Van Gogh are famous artists, paintings are done on paper or canvas, etc. After reading the book, he hopefully fit the new information about murals into his conceptual understanding of art.

Here’s a simplified idea of how some of that information might be organized in his mind. The left and middle boxes represent what he already knew about art, while the box with dotten lines on the right represents new information.

Since he already had some context of what art is, he was able to fit this new art form into his previous understanding of art. As time goes on and he gains more knowledge and exposure to this topic, this understanding will grow and deepen.

If there’s no background knowledge or context for understanding a new fact or idea, the chances of a student learning is pretty slim.

When a student focuses on the surface structure of a problem (like laps around a track), he’s recalling related background knowledge (laps, tracks, runners) to see if it will help to solve it, or to see if there’s anything new that can be attached to what he already knows. If this is how we learn and understand, we can hardly fault a student for placing so much emphasis on the surface structure. Though surface structure can be helpful for absorbing new conceptual information, it wasn’t enough to help my son solve the “laps around the track” addition problem that I presented to him.

How can we help students recognize deep structure, which is also a very important skill, and promote transfer?

The short answer: lots of practice.

In my next post, I’ll go into more depth about how to help transition from surface structure to deep structure.

-Sara McGuinn, Northshore Tutoring Director

Looping: Becoming a Believer!

I started at Nurturing Wisdom before we incorporated looping as a major strategy for our tutoring. Now, I can't imagine doing a session without it, no matter the subject matter!

Looping has transformed me into an even better tutor than I thought possible. Because I'm only giving students one problem at a time, I'm constantly assessing my students as they work. I have to decide if they're showing mastery of a concept and are ready to move on to a harder problem, or if I need to keep repeating that same skill. Now, I'm truly teaching to mastery.

I started using looping with my ACT math students as a way to review a large number of algebra and geometry skills. Thanks to the book Why Don't Students Like School, I'm learning that students need to be exposed to concepts repeatedly in order for those concepts to be stored in long-term memory. I've realized that, when I first started tutoring, I wasn't giving students frequent enough practice.

A breakthrough for me came when I started working with a student who seemed to forget concepts after only one week. I started using looping with her, reviewing the same math concepts each week. Even though she was seeing the same material session after session, she made incredible progress because I was constantly taking these concepts to higher and higher levels. She was able to take her ACT math score from the low teens to a 25 by the time she took her second ACT test!

Before I started using looping, my students' ACT math scores would increase by just a few points. Now I see even bigger jumps! The score increases aren't the most important thing, though. The best part is that I'm seeing students' confidence increase greatly. Looping helps me give my students many chances to be successful with math.

During my three years at Nurturing Wisdom, I've gone from being intrigued by looping, to loving it, to being a true, devoted believer in how powerful of a strategy it can be for tutoring! I'm never going back!

~Heather Roan, Staff Development Coordinator

Surface & Deep Structure

My 7-year-old son and I were watching my older child compete in a track meet. It was a chilly day, and we’d been standing outside, huddled under our coats, for about an hour. As we stood at the fence waiting for the next event to begin, my son sighed, looked up at me and said, “Mom, I’m bored.”

The teacher in me always looks for moments like these to squeeze in some mental activity, so we started talking about the track.

“Do you know how many meters long the track is?” I asked. He shook his head. “It’s 400 meters,” I said. “So when a runner does one lap around the track, he runs 400 meters.”

That seemed to make sense to him, so I figured I’d start off with an easy problem. “All right,” I said, “so if a runner does two laps around the track, how many meters is that?”

I’ll never forget the look of surprise on his expressive face. Shaking his head, his brow knit with genuine confusion, he responded, “What? Mom, I don’t know. We’ve studied meters before in school, but we’ve never studied tracks!”

This time, it was my turn to be surprised. My son was in 2^nd grade, so I knew he had the arithmetic skills necessary to compute 400 + 400. But for some reason, he didn’t recognize that this was just an addition problem. I smiled and crouched down next to him so we could talk through how to get the answer. But in the back of my mind, I was at work on a problem of my own: why was he so baffled by what I thought would be a simple problem?

The answer, I found, lies in the concepts of structure of problems, explained by Willingham in his book Why Don’t Students Like School?

Word problems, such as the one I gave to my son, all have two elements to them: their surface structure and their deep structure. We adults recognize immediately that the ‘laps around the track’ problem is, at its core, an addition problem; that is, its deep structure is an addition problem. The fact that we’re talking about meters on a track does not change what we do to solve the problem. Actually, we could quite easily think of dozens of ways to “disguise” problems with the same deep structure:

• If there are 2 gallons in 1 jug, how many gallons are in 2 jugs?
• If you need 3 eggs to bake 1 cake, how many eggs do you need for 2 cakes?
• If there are 5,280 feet in 1 mile, how many feet are in 2 miles?
• If there are 20 ziggiblops in 1 splutzik, how many ziggiblops are in 2 splutziks?
• If there are 12 # in 1 ^, how many # are in 2 ^?

The “disguise” is the surface structure of the problem. And, as you probably noticed in the last two examples, the surface structure need not even be familiar (or pronounceable) for you to recognize the deep structure and solve the problem.

The disconnect that arose for my son is that while he had the skills and knowledge necessary to solve the problem, he got hung up on the surface structure of the problem – laps around a track, which he had never formally studied – and failed to recognize the deep structure – addition, something he’s been practicing for months on end.

Being able to apply knowledge to new and unfamiliar problems is called transfer, and it’s something that we teachers and parents try to encourage in our kids. But, as you may have noticed with your own kids, knowledge transfer isn’t easy. There are reasons why it’s tough, and there are things we can do to promote knowledge transfer – but we’ll get into those another time. For now, see what you can do identify the surface and deep structures in problems you encounter – and have patience with your kids (or yourself!) during the countless repetitions of problems that will be necessary for successful transfer of knowledge.

-Sara McGuinn, Northshore Tutoring Director