Fourteen years ago I started the Higher Order Thinking Skills (HOTS) project as an alternative approach to teaching Chapter 1 students. The approach was later extended to learning disabled and gifted students. HOTS proved to be highly effective, producing substantial gains in standardized test scores in both reading and math. In addition, 10–15 percent of the Chapter 1 and learning disabled students in the program are currently making honor role, indicating that the gains produce a form of transfer that enables the students to learn classroom content more effectively.

The success of HOTS demonstrated that the major impediment to learning on the part of educationally disadvantaged students was that they did not understand what it meant to understand. We found developing a sense of understanding was a more efficient way of developing academic skills and knowledge than drilling these students in the skills alone.

I then began to wonder whether by spending more time on developing a sense of understanding—and thus less time on direct instruction—we could increase students' achievement in a content area. The HOTS project has spent the past four years developing such a curriculum for middle school mathematics.

## Supermath's Lessons

We chose math as the area to explore because research suggests that the typical student has little interest in math after the 4th grade. Our goals are to increase content achievement by developing students' mathematical understanding and to dramatically reduce the necessity for a great deal of direct practice. The result is

*Supermath*. Preliminary evidence shows that, with a few changes, the instructional and curricular techniques that helped educationally disadvantaged students develop a general sense of understanding in fact are helping all students increase mathematics achievement.Our experience with HOTS and

*Supermath*suggest the following three principles for developing students' understanding in ways that transfer to increased achievement. We must (1) create a learning environment that is consistently intriguing; (2) combine visual and interactive experiences with Socratic forms of conversation that help students create mental models and generalize their experiences; and (3) develop cognitive architecture (a system of thinking activities) that unifies learning experiences.## Intrigue the Students

Learning content that students see as foreign to them requires a great deal of effort on their part. Thus, it is important to first intrigue them. One of the very first lessons in HOTS was to avoid wherever possible trying to intrigue students by tying concepts to real life. So much of the students' real life experiences were difficult and school was a haven away from problems.

Also, in math, very few of the concepts are directly related to students' lives. The following problem, for example, attempts to introduce fractions in a contrived way: If you have 5 people who want to eat 2 slices of pizza, how many parts do you have to divide the pizza into, and what fraction of the pizza will each one eat? No one orders, slices, or eats pizza that way. Such attempts to relate math to real life give the students the message that math is phony.

A more productive approach is to build upon students' sense of fantasy. Elementary and middle school students, having grown up with Big Bird and MTV, have a highly developed sense of fantasy. The goal of a curriculum should be to intrigue students by drawing them into a setting of adventure, mystery, exploration, and a sense of being what they dream of being. HOTS students pretend to be cetologists, aeronauts, and explorers.

*Supermath*students drive a Corvette around San Francisco, order a hamburger on Mars, design a dream home, and appraise gems. The curriculums do not pretend that math is critical to students in their current lives, but they do suggest that math is critical to their imaginings of their future, a much more powerful draw.The middle school years are ones in which students dream and think that many things are possible for them.

*Supermath*is designed around the unfamiliar rather than routine experiences. For example, even though most middle school students aren't familiar with golf, the*Supermath*curriculum uses golf rather than the more familiar basketball as a sports setting. Golf is a noncontact sport that is likely to appeal to girls as well as to boys, an important consideration when attempting to intrigue all students about math. The curriculum arouses their curiosity about golf and inspires them to think about trying to play golf in the real world. If, in addition to learning key math concepts, students decide to expand their horizons and pursue golf outside of school, so much the better.## Help Students Develop a Mental Model

Once students have been intrigued, the next step is to help them develop a mental model of the concept. Few students can imagine what negative numbers, decimals, and the relationship between language and math look like. Without mental models students have a lot of trouble going beyond applying rote rules to manipulating mathematical relationships.

Constructing good mental models is facilitated by using visual and interactive materials. Most current methods, such as using number lines, are simply not sufficiently interactive for students to develop an intuitive sense of the concept. Take, for example, decimal place value. Students memorize rules about decimals and look at number lines, but what does .025 really look like? If students do not understand what decimals look like, they find it difficult to make inferences about the properties of place value.

To help students develop a mental model of decimal place value,

*Supermath*students play a computer game called “Where in the 'Hood is Carmen San Decimal?” where they chase after the crooked Carmen. The addresses of the places they search are in the form of decimals. First they search buildings whose addresses are one-place decimals. When they determine that Carmen has fled, for example, between buildings .2 and .3, they enter the alley to find 9 garbage cans with the addresses .21–.29. When they determine that Carmen has escaped between garbage cans .27 and .28, they enter that alley and find 9 mouse holes with the addresses .271–.279. A search of the mouse holes brings success as the students haul Carmen off to jail. After students have played this game for a while, the notions of place value and the relationship of different types of decimals become intuitive.Another example of a math concept that students never seem to understand intuitively is why subtracting a negative number is the same as adding a positive number. To help students develop a mental model for the addition of signed numbers,

*Supermath*developed a computer simulation called “Three-Smile Island.” The object is to keep a nuclear reactor from exploding by using an injector and vacuum cleaner to bring the radioactive positons and negatons inside the reactor into balance. Students then intuitively see how vacuuming negatons (the mental model for subtracting negative numbers) has the same effect as injecting positons (adding positive numbers). Students later learn to use numbers and signs to represent the process of balancing opposite quantities.One of the biggest needs in the math curriculum was to help students to construct their own mental model of how to convert language to math and vice versa. Such conversion is critical to solving word problems, but is difficult for most students. The solution was to create a new genre of software called “Word Problem Processors.” In this simulation, students communicate with a lonesome space creature inside their computer. Students write stories in the form of word problems to entertain the creature.

Having students write word problems is not new. What is new is that the creature understands English and speaks math. Simulated artificial intelligence techniques enable the creature to judge whether it can or cannot understand the stories and whether the stories are too simple to allow it to speak math. If the creature can understand the story, it provides the solution. If the creature cannot understand the story, it tells the students why, and they then revise their story.

By constantly trying to understand the creature's reactions, students begin to use comprehension strategies to solve problems. Later, when students solve word problems created by one another, they intuitively look first for the underlying structure of the language and think of how the creature would have reacted. The students now have a mental model of how math and language go together and can solve new word problems.

However, as fancy as the technology is, the construction of mental models does not result from the use of the technology. The key component is sophisticated forms of conversation between teachers and students that help students think about the general implications of the activity. Little systematic learning will occur without appropriate forms of interaction between students and teachers. It is critical that teachers ask the right questions and consistently probe the students' answers for understanding. Without such questioning, the game remains a fun experience as opposed to a means to develop a mental model.

Both HOTS and

*Supermath*use Socratic instruction techniques: that is, the teacher teaches by asking questions instead of by telling. For example, even with the “Word Problem Processors,” teachers must constantly ask students why the creature reacted to their language the way it did and why the creature solved the problem the way it did.While the initial key questions are contained in the curriculum, the most important part of the conversation is the follow-up probing of the students' initial, usually incomplete, answers. Without consistent probing, even the most creative curriculum results in dogmatic forms of teaching, and hands-on activities are converted into applying dictated rules.

## Create a Cognitive Architecture

A third requirement for developing understanding is to build all activities around a set of thinking strategies. The two most important thinking skills to develop are metacognition and decontextualization. Metacognition is the ability to systematically formulate strategies; decontextualization is a fancy word for generalization—the ability to apply a concept learned in one context in another context.

To stimulate metacognition, HOTS teachers use software in an indirect way. Ignoring what the software is designed to teach, they have students try to figure out how to use the software. This creates interesting dilemmas for the students, who are then urged to articulate and discover strategies for solving the dilemmas and then to discuss the success of these strategies. I call using software as an opportunity to have thoughtful discussions “learning dramas” (Pogrow 1990a, 1990b). The HOTS computer activities provide a setting around which reflection and conversation can occur. This reflection about the use of language, both textual and oral, produces substantial gains in reading.

The

*Supermath*curriculum goes a step further and choreographs the computer-based situations so that the only way to resolve the dilemmas is to use math. For example, in the golf program, the students must enter angles and unit lengths to drive the ball into the hole. Once students have figured out the angle at which to hit the ball, they discover they cannot get the ball into the hole by using whole unit lengths. The only way to resolve the dilemma is to use decimals. As students improve their scores, they are still unable to beat par by simply estimating angles and distances. To get a hole in one, they need to use measuring tools. This leads to a discussion of the pros and cons of estimation versus measurement. Converting the tic marks on rulers to decimals requires the use of proportion. As a result, the concepts of angles, estimation, decimals, proportion, and measurement are integrated in the students' minds.Students develop decontextualization thinking skills by linking key concepts across different software settings. For example, after they understand the concept of angles in the golf game, they use their new understanding to navigate a ship in the

*Voyage of the Mimi*, science software about a sailing adventure. They also discuss angles when they design a home using an architecture program. The curriculum is designed to give students enough experiences in relating concepts in different contexts to be able to begin to think about ideas in general terms.## Implement the Principles for Generating Understanding

While the key principles for generating understanding are simple, using them consistently and well is complex. Many of the widely advocated approaches to reform simply do not measure up. For example, typical manipulatives currently used in math often are not interactive, and they fail to intrigue older students.

While computers offer tremendous potential for stimulating learning, often the curricular approaches for using them are not effective. Integrated learning systems are expensive electronic versions of traditional ways of teaching. Simulations and tools offer the potential for new approaches to teaching, but teachers need curriculum and pedagogical strategies to use them.

Both HOTS and

*Supermath*combine technology, masterful teaching, and a creative curriculum. But educators must wonder: Is it really feasible to intrigue students, introduce dilemmas, help them develop a mental model, and talk to them about their model? When are students going to learn content? How do we provide time for all the activities central to developing understanding? The answer: Once HOTS students had a sense of understanding, they were able to learn what was taught in the classroom the first time it was taught. As a result, knowledge, school performance, and test scores increased.The same thing seems to be happening with

*Supermath*. Results from the first year of pilot classes indicate that the time spent developing understanding is increasing students' math achievement. A complete two-year general math/pre-algebra curriculum,*Supermath*requires students to use some traditional forms of practice. Students, however, infer most of the concepts before they practice the skills. While more research will be needed to determine the optimal mix of discovery and direct practice, it seems clear that students who develop a sense of understanding need less direct practice.The combination of greater understanding and being taught in an intriguing way increases students' interest in content. Teachers report that students often say, “Are you sure that this is math? This is too much fun!” At first most students are uncomfortable with having to express and explain their ideas. Their teachers report amazement about how primitive student conceptions of math are—even for topics that have been covered for many years. Students adapt relatively quickly, however. Some students who had just taken their standardized math tests said the dreaded sections on signed numbers and word problems were “easy.”

## Helping the Educationally Disadvantaged

There is no doubt that the educationally disadvantaged profit from strategies that teach them to understand. With this preparation they have a level playing field in a content course that emphasizes problem solving and the expression of ideas. Chapter 1 and learning disabled students who were inadvertently placed into the more sophisticated

*Supermath*without prior HOTS experience had tremendous problems. Those with one to two years of HOTS experience, however, did well in both homogeneous and heterogeneous*Supermath*classes. We recommend that disadvantaged students not be placed into*Supermath*until they have had at least a year of HOTS or some other general thinking skills program.Teachers report that although students who were formerly classified Chapter 1 or learning disabled sometimes had less initial math knowledge, they held their own in class discussions and activities. These results seem to indicate that it may be possible to design effective strategies for eliminating tracking for most educationally disadvantaged students.

## Where Do We Go from Here?

Educators have consistently underestimated the importance and complexity of developing a sense of understanding in students. It is possible to design more powerful curriculums that enable students to gain this sense of understanding, however.

Just as educationally disadvantaged students usually do not understand how to deal with ideas in general, most average students in math do not seem to understand how to think mathematically. If we can apply the basic principles that underlie the development of a sense of understanding and motivation for a wide range of students, then the time these strategies take away from direct instruction of content will be well spent.