Supermath: An Alternative Approach to Improving Math Performance in Grades 4 Through 9

Stanley Pogrow shares a decade's worth of experience in developing an approach to teaching math to upper-elementary and middle school students that simultaneously increases basic skills, improves problem solving, raises test scores, and sparks student interest in math. For details, read on.

By Stanley Pogrow

MY WORK over the past 24 years has focused on using technology to produce more sophisticated forms of learning after the third grade. This work started with the Higher Order Thinking Skills (HOTS) program for Title I and learning-disabled students. (For more information on the HOTS program, visit www.hots.org.) We used lessons learned from this pioneering effort as we developed Supermath, an alternative approach to teaching math to all students. Supermath provides interesting opportunities for districts to better meet the accountability challenges of No Child Left Behind (NCLB) while also producing reflective mathematics learning as celebrated in the November 2003 issue of the Kappan.

Background of Supermath

In the early 1990s, the National Science Foundation (NSF) funded the development of Supermath as part of an effort to produce new and innovative approaches to and materials for teaching mathematics. In designing Supermath, we sought to develop an approach that would simultaneously increase students' basic skills, problem-solving ability, test scores, and interest in math. We aimed to achieve these results for both advantaged and disadvantaged students and to be far more successful in doing so than conventional approaches, even those that used technology, had been. Given these ambitious goals, it was clear that the Supermath materials would have to be creative and engaging and would have to incorporate and systematize powerful curricular and pedagogical techniques.

The use of technology to improve learning has traditionally focused on presenting content in an individualized fashion and then testing students. Fancy graphics and animation are used to intrigue the students. Technology serves to automate instruction rather than to create new forms of instruction. While this approach has value, it generally produces gains only in the early grades -- and primarily on the program vendor's test. There are more creative simulations and tools available on which to build new approaches to instruction. But, while valuable and able to provide for exploratory learning, these technologies tend to be hit or miss and place heavy burdens on teachers to figure out how to use them effectively. It is also hard to demonstrate the impact that these simulations and tools have on improved content skills and test scores.

HOTS demonstrated that it was possible to use technology systematically in ways that produced progressive outcomes -- comprehension and self-concept -- while also producing even greater gains on traditional achievement measures, such as test scores. The primary lesson from HOTS is that developing a more powerful approach to learning requires insight into what is inhibiting learning and ideas about how to overcome these inhibitors. Then, and only then, should one think about using technology to improve instruction and student performance.

In the case of HOTS, I discovered that for disadvantaged students the key inhibitor of reading comprehension and general learning was not understanding "understanding," a problem that could be addressed only through intensive conversations about ideas.1 Based on this insight, HOTS uses technology to provide students a common experience that serves as the basis for shared conversations, much as a family uses life experience as the basis for dinner-table conversation. In this approach to technology use, curriculum and training were designed to generate an environment conducive to sophisticated and intensive Socratic conversation and to maintain it over the one- to two-year period needed to develop students' sense of understanding. Using technology indirectly to promote conversation, as a table setter so to speak, worked powerfully.

Could an indirect use of technology also work to develop specific content skills and increase student interest? In order to identify the inhibitors that prevent learning math, we conducted extensive conversations with teachers and students. These conversations, combined with a literature review, led us, not surprisingly, to conclude that the key inhibitors for the average student were 1) not understanding many of the concepts, 2) seeing no purpose for the concepts, and 3) seeing math as a series of arbitrary, unintelligible rules imposed by adults. Teachers lamented the difficulty of explaining fundamental math concepts in ways that both made sense to most students and sparked their interest.

Our conclusions about how to solve these problems generally mirrored the reform literature of the 1990s, which said that the teaching of math should be more exploratory, should use pedagogical techniques such as constructivism that enable students to reflect on math principles, and should focus on solving problems. But how could this be done systematically and effectively on a larger scale?

Description Of Supermath

Supermath differs from conventional reform efforts in the way it actually implements complex forms of instruction and in the types of problems it asks students to solve. The prevalent reform wisdom advocates "authentic" math problems, i.e., those connected to the real-world experience of students. Unfortunately, while math is fundamental and critical, younger students can get along quite well in real life without using it, except for perhaps making change and shopping. (John Marshall does an excellent job of describing the difficulties in tying math instruction to meaningful real-world examples.2) While adults are required to use math in real-life situations, teaching math to upper-elementary and middle school students by using situations adults encounter is very problematic.

Students who are turned off by math are also likely to be rebelling against the notion of adulthood -- or at least the notion of being like most of the adults they know. Such students think that they are different, immortal, and can easily achieve anything they want. The worst thing that you can do to students who think that math is a set of pointless, adult-imposed rules is to tell them that they will understand the need for math when they grow up or that learning math will make them more successful adults.

It became clear to us that to draw students into the world of mathematics, the applications and approach of Supermath had to be based on the types of experiences and modes of learning and thinking that they valued. This did not mean that the rigor of the mathematics had to be compromised, but rather that the mode of presentation and the types of problems had to correspond to what students cared about and how they explored new ideas. The key was to understand how kids thought about things. Teachers repeatedly complained that their students lived in a fantasy world and spent most of their time seeking entertainment. This problem provided us with the basis for a solution, namely that all Supermath applications and exercises would be built on fantasy.

In Supermath, a problem is considered authentic only if students view it as important to think about -- on their terms. All the Supermath concepts are built around imaginary scenarios. For example, in one Supermath unit students play the role of international jewel detectives, in another that of police searching for the criminal mastermind Carmen San Decimal, the most evil decimal in the world. In other units students are race car drivers, astronauts, athletes, controllers of a nuclear reactor, space travelers communicating with aliens, and so on. (Juvenile humor is used throughout.)

Supermath does not present mathematics concepts directly, but rather provides fantasy settings in which students discover and master these concepts on their own. Indeed, it is hard to tell by just looking at the software how it is tied to math at all, let alone what specific math concept is being taught.

How is the formal math content brought in? The program is designed for students to begin playing very quickly. Then, just when they get interested and start feeling confident, obstacles are introduced -- e.g., the evil decimal escapes between buildings, the car that the student designed won't go fast enough, or the student can't lower his or her golf score enough. At this point the students go from feeling that they have mastered the game to feeling frustrated at the unexpected hurdle.

This orchestrated frustration provides an opportunity to introduce math principles and skills because the settings are designed so that the only way to resolve the dilemmas is by applying math. For example, to make the car go substantially faster, the students must understand the ratio and decimal information on the system performance printout, and to improve the golf score, they must switch from estimating unit lengths and angles to actually measuring them and from using whole numbers to using fractions and decimals (it also helps if you know about positive and negative numbers).3

Math saves the day by giving students the means to overcome the dilemmas and master the setting. The students absorb ideas quickly and are eager to apply them to the game. Teachers are provided with a curriculum that shows them how to bring the math to bear quickly and efficiently. In addition, teachers are trained to interact with students in ways that allow them to learn the concepts within an exploratory process.

The Supermath curriculum creates the settings, dilemmas, and math-based resolutions to its games using the same techniques as those used by writers of novels and plays. Each unit is an episode with a story line that depends on mathematics. This indirect approach to using technology to teach subject matter creates situations called "Learning Dramas."

In Supermath, all math concepts are learned and used within a context where they are needed to reach a valued end. Indeed, this is how adults use math in the real world. The average adult does not go into the office and use math for the sake of using math; it is incorporated into work to resolve problems critical to a desired outcome. Indeed, the application of mathematics in the fantasy settings is as real to the students as the real-world use of math is to adults.

Using technology to enable learning indirectly and envisioning settings and dilemmas that will both interest students and support the comprehensive coverage of math skills and problem-solving techniques is difficult and time-consuming. At the same time, this approach to using technology to teach math skills ultimately has a higher payoff than conventional approaches. Our research shows that Supermath improves both retention and test performance and increases expressed interest in math. John Bransford at the University of Washington similarly found that using technology indirectly to create a context that would stimulate a discussion of physics principles produced greater learning than using technology to present the principles directly.4

Supermath in the Hood

Fortunately, we were able to develop a comprehensive set of units for Supermath that use technology in this indirect way to develop students' math proficiency and interest. Supermath's Learning Dramas make it easier and more practical to incorporate advanced teaching and learning techniques. For example, it becomes possible to teach in a way that enables all students to create mental models that will allow them to infer and extend mathematical concepts. Concepts that were formerly arcane and boring become intuitive and engaging.

One example of this technique of mental modeling is the Supermath approach to learning how to compare numbers that include decimals. If you ask students which number is larger -- 3.2 or 3.19999 -- most will say 3.19999 and yawn. It is counterintuitive to suggest otherwise. And most teachers lack the tools to really convince students that the counterintuitive answer of 3.2 is actually correct. Teachers can tell students the correct answer and explain its corresponding rule, and most students will follow it, but this is good behavior, not mathematics learning or mathematical reasoning. Chances are that the rule will quickly be forgotten or, if remembered, will not be extended to other problem-solving situations.

However, the Supermath setting enables students to easily infer in an intuitive fashion the role of decimal places and the rules for determining the larger of two numbers with decimals. This is because the setting is designed to enable students to form a picture in their mind of what a decimal is and how it behaves and to use that image as a mental model for math inference.

In the decimal example, the Supermath setting is a "hood" with a number of different levels of structures. There are houses, garbage cans in alleys between the houses, and mouse holes between the garbage cans. The various structures have different numbers of decimal places in their addresses. At each level, students are told that as they go to the right of the screen they are going uptown and that the addresses are getting bigger. (Yes, we have piloted this with students who live in remote rural areas where there is no downtown, and they adapt quickly.) The students quickly discover that the evil decimal likes to escape from one level to the next lower level to hide (kind of like Saddam). As students narrow the search by moving from one level of the hood to the next, more precise, level, the addresses contain an additional decimal place. Students eventually intuit on their own that 3.19999 is in the alley to the left of the 3.2 building and is therefore smaller.

In traditional math teaching, the teacher might draw a number line on the board and try to illustrate the same idea. Unfortunately, the number line means nothing to many students, and the well-intentioned teacher is therefore using a strange artifact to teach a nonintuitive concept. In the game-like Supermath setting, the same basic idea is put into a form that is very intuitive to students. The hood is actually a mental model for a three-dimensional number line that can be manipulated and that can give feedback. Students using Supermath are able to think about the mathematical concepts within the context of the game without initially realizing that they are engaged in math reasoning. However, with sufficient experience playing the game and with the related classroom discussions outlined in the curriculum, students begin to develop a mental model to make inferences about the properties of decimals outside the hood setting. The students begin to see patterns in their answers that allow them to infer fundamental rules for using place value to determine the size of decimals and to understand the practical significance of adding more decimal places to a number (i.e., greater precision). Through the game, mathematics is transformed from a subject with adult-imposed rules to a game-like process in which the students can discover many of the rules, skills, and concepts we want them to learn. By making the rules their own, students retain them better and are able to apply the skills in a variety of problem-solving contexts, which increases their test scores and their interest in math.

Word Problem Processors

Another example of Supermath's ability to apply advanced teaching techniques is its approach to the most difficult of all pre-algebra topics -- word problems. Teaching students to solve word problems is so difficult because it involves the interaction of two symbol systems -- language and math -- and many of the students are weak in one or both systems. (Math teachers have always known that the biggest impediment to learning math for many students is poor language skills.) Indeed, the challenge of teaching word problems is so difficult that it demands an advanced, constructivist approach.

In order to support a constructivist approach to teaching students to solve word problems, Supermath created a new genre of software called Word Problem Processors (WPP). With WPPs, students write stories to a lost and lonesome space creature whose disk ship has crash-landed inside their computer. The creature is stuck there and wants to be told a story. WPPs use artificial intelligence to enable the imaginary creature to parse students' language and determine if the language/story makes sense. If not, the creature gives students feedback about what is wrong with their stories (students can choose to receive the creature's feedback in Spanish or English). Students have to use this feedback to revise the language in their stories before they can get any further reaction from the creature. When a student writes a story in the correct language and a math calculation is involved, the creature produces a step-by-step solution.

The role of the teacher is to look at students' portfolios of stories and solutions and act befuddled: "Why did the creature react mathematically to this story and words and react in this different way to your other story?" This metacognitive prompt causes students to reflect on the creature's responses. Such prompts are an essential part of channeling students' exploration into systematic learning. However, this process converts the teacher from a rule-giver and referee into a coach and converts the students from story-problem solvers into sociologists investigating the behavior of the space creature. It frees both from previously dreaded (if truth be told) roles and turns them into explorers.

There are seven different levels of WPPs, ranging from simple one-step problems to complex equations.5 The following is an illustration of the process for the simplest WPP level -- one-step story problems -- though the process is the same for all the levels. At this level, students create a story with three sentences labeled Open, Change, and Question. Students easily construct each sentence by choosing from a menu of sentence fragments. For example, a student might write the following story: Joan bought 3 magazines. Jose 2 birds. How many magazines did Joan and Jose obtain left over? She would then ask the creature to react, and the creature would respond: You need to add the Change Verb.

The student would then figure out that she needs to add a verb to the second sentence, modifying the story to read: Joan bought 3 magazines. Jose found 2 birds. How many magazines did Joan and Jose obtain left over?

The creature would then react by responding: You need a different extra word in the Question.

This feedback indicates that the "extra word" fragment in the third sentence needs to be changed. So the student could then modify the story to read: Joan bought 3 magazines. Jose found 2 birds. How many magazines did Joan and Jose obtain altogether?

This time the creature "understands" the story, but responds, You have insulted my intelligence! This story is too easy. They bought 3 magazines. You gave the answer in the opening statement. Please give me a chance to calculate.

Now the student would have to come up with a way to change the story so that the creature has to do a calculation.6 For example, she might change the "object" in the second sentence so that the story now reads: Joan bought 3 magazines. Jose found 2 magazines. How many magazines did Joan and Jose obtain altogether?

The creature would respond: You created a story that I understand and like.

Then the creature would provide the step-by-step solution and the answer, five magazines. After completing the process, students can store their valid stories and solutions. Because English is such a complicated language, even at this basic level, the program has over 1,000 grammatical rules built in. At the higher levels, the mathematics is far more complex, with many steps in the solution.

The WPP process does not pressure students to come up with solutions to the stories; instead, teachers focus on getting students to explain the creature's reactions and solutions. With sufficient WPP experience, students begin to understand how the space creature thinks about language and the nature of its mathematical reactions. After five or six days of such practice, students can see in their minds how the creature responds mathematically to language at a given level of difficulty.

Students engage in WPPs in teams. After five or six days, each team has a portfolio of saved stories and solutions. Then the students can play a competition organized by the computer around each team's bank of stories. When students are challenged to solve another team's stories, they do so by constructing in their heads how the creature would react to the problems -- which then becomes how they react.

In order to make the team competition interesting, students can include distractors in their stories and can alter the structure of the language to fool the other teams.7 When they do so, the teacher should again act befuddled and ask the metacognitive question: "Why does the creature think this story is the same, even though the language is different?"

For example, to fool the others, a team could first add a distractor to the magazine story, changing it to: Magazines cost $1.25. Joan bought 3 magazines. Jose found 2 magazines. How many magazines did Joan and Jose obtain altogether?

Then to further disguise the real problem, the team could change the language structure to read: How many magazines did Joan and Jose obtain if magazines cost $1.25 and Joan bought 3 magazines and Jose found 2 magazines?

Of course the other teams are all doing the same things. Thus students experience a type of Play-Doh communication process made possible by artificial intelligence, in which they manipulate and puzzle about the mathematical outcomes caused by variations in the language they use. By seeing the effects of their language choices on the creature's mathematical reactions (solutions), the students construct their own understanding of how language and math interact and are able to solve word problems intuitively. Indeed, when we test students who have solved WPPs with story problems of equal mathematical complexity but different language structures, they are able to solve the different forms. In other words, students are able to generalize their constructivist experience in linking language and math to new types of problems, which is a breakthrough. Even disadvantaged students can come to enjoy solving word problems once they view the activity as a sociological inquiry into the behavior of the creature.

The Supermath WPP units may solve one of the biggest problems in teaching pre-algebra to most students. Solving word problems is a fundamental skill, one that is required by an increasing proportion of state tests. When disadvantaged students who have experience solving WPPs come out of a test, they report that the easiest part of the test was the word problems and that the distractors on the test were ridiculously lame and easy to spot.

Tuning In to Kids

The power of the Learning Drama approach is a function of the extent to which students are intrigued by the fantasy challenges. Those who think that these programs have to include super-fancy graphics should look at the primitive graphics used in "South Park," "The Simpsons," etc. The best graphics are the graphics of the mind, and those are stimulated by interest. This means that using technology effectively to teach math requires not only an understanding of math but a sense of what interests kids and how they think about life and ideas.

This vision of curriculum, teaching, and technology use requires a great deal of creativity/weirdness. Over time Supermath has found ways to tap into kids' interests and views to develop novel, creative, and effective dramas that teach most of the difficult and nonintuitive pre-algebra concepts.8 Whether students are discovering the principles of rounding numbers by figuring out if the cashiers at the local McDonald's on Mars are trying to rip them off, or playing golf to learn about angle estimation and measurement, or designing and testing a car to learn systems analysis, those who previously were uninterested and doing poorly in math suddenly are not only learning skills but are thinking mathematically -- and thinking that math is cool. Simply stated, Supermath is a curriculum that presents adult ideas from a kid's point of view in ways that affect students cognitively and emotionally.

By providing a comprehensive set of 22 creative supplemental units within a consistent developmental framework and detailed teacher guides, Supermath provides the needed intensity of experience to change the way students think mathematically. These changes simultaneously increase reflectiveness and interest, while also increasing skill acquisition and retention. Hence test scores increase.

Implications for Practice

While Supermath does not replace the need for a coherent and aligned math curriculum, it breaks down the walls between progressive teaching techniques, such as constructivism, and traditional approaches to teaching and learning, making a seamless whole. Schools and districts can easily integrate the use of Supermath units and conventional skills instruction into ongoing efforts to meet state standards and student needs. Once students complete a Supermath unit, they can practice and automate the relevant skills using conventional materials. However, the students would require significantly less practice than otherwise to automate the skills and would have the ability to apply the learned skills to a wider range of problems and situations.

Our research shows that the combination of Supermath and conventional curriculum materials produces much more growth than using traditional methods alone. In addition, research based on the extensive piloting of Supermath has generated recommendations for balancing its use with traditional materials for a wide variety of program applications -- from remediation with Title I students to enrichment with gifted students, and from inclusion in regular instruction to use in summer and after-school programs.9

For teachers, the unconventional process of evaluating or learning to teach a Supermath unit may seem strange at first. But once they are trained to use a few units, the approach becomes second nature. My experience has been that when teachers of advantaged students first come to understand how the program works, they are convinced of its value, while teachers of disadvantaged students are often initially skeptical that their students will or can think mathematically, as called for by the curriculum. Some have even gone so far as to say, "Our kids do not think mathematically." And that is the power of Supermath -- to change perceptions and possibilities by changing the reality of what students actually accomplish. If teachers trust the curriculum and implement it correctly, they are amazed at what their students are capable of and interested in doing. Students, in turn, are amazed to discover that they can figure out and enjoy math.

The natural tendency of schools will be to meet the NCLB math requirements by cramming in as much test prep/content drill as possible. While this strategy may initially increase test scores, the gains will quickly plateau. This approach will also produce another generation of math haters and mathaphobes.

Supermath represents a new curricular approach that combines the best of traditional and progressive techniques in a data-driven balance that enhances both. It shows that it is possible to design a progressive approach that can substantially raise test scores while also transforming how teachers and students come to view the process and value of learning mathematics. If it can be done in mathematics, it can probably be done in other content areas. What we need are more creative, powerful, and synergistic forms of curricula.

1. See, for example, Stanley Pogrow, "The Missing Element in Reducing the Learning Gap: Eliminating the 'Blank Stare,'" Teachers College Record, October 2004, available at www.tcrecord.org/Content.asp?contentID=11381; and Stanley Pogrow, "Challenging At-Risk Students: Findings from the HOTS Program," Phi Delta Kappan, January 1990, pp. 389-97.

2. John Marshall, "Math Wars: Taking Sides," Phi Delta Kappan, November 2003, pp. 193-200.

3. As an illustration of how challenging the Supermath environments are, one of the math educators reviewing Supermath for the NSF became incensed at the golf application. The reviewer felt that it was not an "authentic" application, since golfers do not really calculate angles. She was also upset because she could not get the ball in the hole and therefore felt that the software was not working. Had she bothered to read the curriculum, she would have realized that the game was designed to reinforce the use of decimals, so the ball would not go in the hole unless the player used decimal length. Other reviewers felt that it was unfair to use a golf setting with minority students and that we should have used a more familiar game such as basketball.

4. The Cognition and Technology Group at Vanderbilt, The Jasper Project: Lessons in Curriculum, Instruction, Assessment, and Professional Development (Hillsdale, N.J.: Erlbaum Associates, 1997).

5. The different levels of word problems are: one-step, simple proportion, proportion with conversion, multi-step, graphing vertical motion, graphing horizontal motion, and solving stories that involve one-variable equations.

6. Once students begin creating complex, multi-step stories, the creature gets insulted if they write one that involves only one calculation. The sophistication of the creature's expectations evolves as the level of difficulty in the problems increases.

7. It is important for students to learn to generalize solutions to different forms of language. Research has shown that it is generally possible to get students to solve a particular form of word problem, but they become confused when the language changes, even if the problem is the same mathematically.

8. For a complete listing of Supermath units and skills, contact the author at stanpogrow@att.net.

9. For example, our pilot studies have shown that a good combination for improving test scores seems to be about 40% to 50% Supermath and 50% to 60% traditional instruction for disadvantaged and underperforming students, while a lower percentage of Supermath activities are needed for advantaged students.

STANLEY POGROW is a research professor of educational leadership at San Francisco State University. He wrote this article while serving as the William Allen Endowed Chair and distinguished visiting professor of educational leadership at Seattle University. He specializes in school reform policy and the educational application of technology. Contact him at stanpogrow@att.net for more information on Supermath.