Supermath:An Alternative Approach to Increasing Math Performance in Grades 4-9
by Dr. Stanley Pogrow. To be published in Phi Delta Kappan2004-05
Dr. Stanley Pogrow is currently the William Allen Endowed Chair and distinguished visiting professor of Educational Leadership at Seattle University, while he is on leave from the University of Arizona. Dr. Pogrow specializes in school reform policy and the application of technology. For more information on Supermath the author can be reached at: firstname.lastname@example.org.
"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 LD students. We used lessons learned from this pioneering effort to develop an alternative approach to teach math for all students, called Supermath. Supermath provides interesting opportunities for districts to better meet the accountability challenges of No Child Left Behind, while also producing the types of reflective mathematics learning celebrated in the November 2003 issue of the Kappan." - Dr. Stanley Pogrow
BACKGROUND OF SUPERMATH
Supermath development was funded in the early 90's by the National Science Foundation as an effort to produce new, and innovative approaches and materials for teaching mathematics. The design goals of Supermath were to develop an approach which would simultaneously increase student's skills, problem solving ability, test scores, and interest in math for both advantaged and disadvantaged students--to a far greater extent than conventional approaches, even those that used technology. Given these ambitious goals it was clear that the materials would have to be creative, engaging, and incorporate and systematize powerful curricular and pedagogical techniques.
Using technology to improve content learning has traditionally focused on using the technology to present the content in an individualized fashion and to then test students. Fancy graphics and animations are used to intrigue the students. This approach seeks to automate instruction as opposed to using technology to create new forms of instruction. While this approach has value, gains are generally produced only at the early grades. There are also more creative simulations and tools available that creative instruction can be built around. However, while these are valuable and do provide for exploratory learning, they tend to be hit and miss and place heavy burdens on the teachers to figure out how to use them effectively. It is also hard to demonstrate impact from simulations and tools on improved content skills and test scores.
HOTS demonstrated that it was possible to use technology in ways that produced progressive outcomes, i.e., comprehension, and self-concept, while also producing even greater gains on traditional measures, i.e., test scores. The primary lesson from HOTS is that developing a more powerful approach requires starting with a better vision and insight as to what is inhibiting learning and how to overcome these inhibitors. Then and only then should one think about how to use technology. In the case of HOTS this insight was that the key inhibitor of reading comprehension and general learning was that disadvantaged students did not understand "understanding", and that the only way to develop a sense of understanding was through intensive conversation about ideas. As a result, the approach developed for using technology is best described as "a metaphor for life". In this approach technology is used to provide a common communal experience that serves as the basis for shared conversation, much as a family uses life experience as the basis for dinnertable conversation. Curriculum and training was developed around this approach to technology use to generate a sophisticated and intensive Socratic conversation environment, and to maintain it over the extended period of 1-2 years needed to develop a sense of understanding. This indirect use of technology, as a table setter so to speak, worked powerfully.
Could an indirect use of technology as a table setter also work for developing specific content skills and interest? In order to understand the inhibitors for learning math, extensive conversations were conducted with teachers and students. These conversations, along with a review of the literature, led to the not surprising conclusions that key inhibitors of learning math for the average student were that they: a) did not understand many of the concepts, b) saw no purpose for the concepts, and c) saw math as a series of arbitrary unintelligible rules imposed by adults. Teachers lamented about the difficulty of explaining many of the fundamental math concepts in ways that made sense to most students or interested them. In most ways our conclusions about how to solve these problems mirrored the reform literature of the 90's, i.e., that: a) math should be more exploratory, b) advanced pedagogical techniques such as constructivism should be employed to enable students to reflect about math principles, and c) the learning of skills should focus around solving problems. However, Supermath incorporated two major departures about how to do such reform.
DESCRIPTION OF SUPERMATH
The primary departure of Supermath from conventional reform efforts was about how to actually implement the complex forms of advocated instruction and the types of problems students should be asked to solve. Prevalent reform wisdom was that math problems should be "authentic", i.e., tied to the real world. Unfortunately, while math is fundamental and critical, younger students can get along quite well in real life without using math except for perhaps making change and shopping. (The problem of tying the types of advocated reform teaching to meaningful real world examples was hinted at in the excellent article by John Marshall in the November 2003 issue of the Kappan.[i]) While adults do have real-life experiences that require math, teaching math to upper elementary and middle school students by using problems from the world of adults is very problematic. Students who are turned off by math are also likely to be rebelling against the notion of adulthood, or at least of being like most of the adults around them. Such students think that they are different, immortal, and can easily achieve anything they want. The worst thing that you can do to kids who think that math is a pointless extension of adult imposed rules is tell them that they will understand why they need to do this math when they become adults, or that this is something that will make them a more successful adult.
It became clear that to draw students into the world of mathematics, the applications and approach had to be based on those 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. What it did mean was that the mode of presentation and types of problems had to make sense in terms of what kids cared about and how they explored new ideas. The key was to understand how kids thought about things. Teachers repeatedly complained about how their students lived in a world of fantasy and spent most of their time seeking entertainment. That problem also provided a basis for a solution.
As a result, all Supermath applications and problems are built around fantasy. An authentic problem is viewed as one that kids buy into as being important to think about--on their terms. All the Supermath concepts are built around imaginary settings. For example, in one Supermath unit students are international jewel detectives, in another they are police searching for the criminal mastermind Carmen San Decimal who is 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, etc. (Juvenile humor is used throughout.) Within this overall vision, technology is just used to present the fantasy settings. In the indirect approach, the computer does not present any mathematics. Indeed, if you look at the software you have no idea what math concept it is being used for, or even how it is tied to math.
How is the formal math content brought in? The software is designed so that students can begin playing very quickly. Then just when they get confident and interested, key dilemmas emerge. For example, the evil decimal escapes between the building, or they cannot get the car they designed to go fast enough, or their golf score low enough, to get a good grade. Dramatic techniques are used to introduce dilemmas into these setting that are obstacles that need to be overcome for successful mastery of the fantasy game or task.
At this point the students have gone from confidence that they have mastered the game to being frustrated at the unexpected hurdle. The setting is designed so that the only way to resolve the dilemmas is to infer math principles or use key math skills. For example, making the car go substantially faster requires understanding the ratio and decimal information on the system performance printout, and improving the golf score requires switching from estimating unit lengths and angles to measuring them and switching from whole numbers to fractions and decimals (it also helps if you know about signed numbers).[ii] This orchestrated frustration now provides an opportunity to introduce the needed math concepts and skills to an eager audience. They absorb ideas quickly and are eager to apply them. Teachers are provided with a curriculum that shows them how to bring the math to bear quickly and efficiently. Math saves the day by giving students the means to overcome the dilemma and master the setting. In addition, training is provided to teachers on how to interact with students so that the concepts are learned within an exploratory process.
The Supermath curriculum uses the techniques of writing novels and plays to create the settings, dilemmas, and math-based resolution process. Each unit is an episode that has a story line that creates a context in which mathematics is absolutely essential and in which mathematics comes to the rescue. This indirect approach to using technology is called "Learning dramas".
As a result, all math concepts are learned within a context where there is a need for their use 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. Math is incorporated into work to resolve problems critical to a desired outcome. Indeed, the application of the mathematics in the fantasy settings is as real to the students as the real world use is to adults.
Clearly, using technology as an indirect enabler, and envisioning enough settings that will both interest students and support dilemmas for incorporating a comprehensive coverage of math skills and problem solving experiences, is difficult and time consuming. At the same time, this approach to using technology and math skills ultimately has a higher payoff than conventional approaches. Our research shows that both retention and test performance increases, as does expressed interest in math. Our research corroborates experimental findings by John Bransford at the University of Washington. He showed that using technology indirectly to create a context to stimulate a discussion of principles of physics produced greater learning than using technology to present the principles directly.
Fortunately, Supermath was able to develop a comprehensive set of units using this more advanced approach to incorporating technology and developing math proficiency and interest. Supermath’s Learning Drama approach makes it easier and practical to incorporate more advanced teaching and learning techniques. For example, it becomes possible to teach in a way that enables all students to create mental models as a basis for their inferring and extending mathematical concepts. Concepts that were formerly arcane and boring become ones that are intuitive and that students want to think about. One example of the technique of mental modeling is the approach used in Supermath for learning how to compare decimals. If you ask students which decimal is larger -- 3.2 or 3.19999, most students are going to say 3.199999 and yawn. It is counter-intuitive to suggest anything else. And teachers really do not have any tools to really convince students that the counter-intuitive answer of 3.2 is actually correct. The teachers tell them what is correct and what the rule is, 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, and that even if remembered, the ideas will not be extended to problem solving situations.
However, the Supermath setting enables students to easily infer the role of decimal places, and infer the rules for determining which of two decimals is larger in an intuitive fashion. This becomes possible because the setting is constructed in such a way that enables students to form a picture in their mind of what a decimals is and how it behaves, and use that as a mental model for math inference.
In this case the computer setting is a “…hood” with a number of levels of structures. There are houses, garbage cans in alleys between the houses, and mouse holes between the garbage cans. The different types of structures have different numbers of decimals in their addresses. At each level students are told that as they go to the right of the screen they are going uptown and 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 Sadaam). As students move from one level of the hood to the next more precise level to narrow the search, the addresses contain an additional decimal point. Students then 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 to try and illustrate the same idea. Unfortunately, the number line means nothing to students and the well intentioned teacher is therefore using a strange artifact to teach a non-intuitive concept. In the gamelike Supermath setting the hood is actually a mental model for a three dimensional number line, and the same basic idea is now in a form that is very intuitive to students and that they can manipulate and get feedback. Students are now able to think about the mathematical ideas within the context of the game. They do not initially realize that they are engaged in math reasoning. However, with sufficient experience and with the related discussions choreographed by the curriculum they become able to use the game as a mental model to make inferences about the properties of decimals outside the setting. With repeated practice at determining which of two addresses (i.e., decimals) are larger using the mental model, students can then find patterns in their answers for inferring fundamental rules of: a) how to use place value to determine size of decimals, and b) what the practical significance is of adding more decimal places (i.e., greater precision). Mathematics transitions from being a subject with adult imposed rules to a game-like process wherein they figure out on their own many of the rules, skills, and concepts we want them to learnwhich become their rules. They are then able to retain and apply the skills in a variety of problem solving contexts and test scores increase, as well as their interest in math.
Another example of the use of advanced teaching techniques is Supermath's approach to teaching the most difficult of all pre-Algebra topicssolving word problems. It is so difficult to formally teach teaching students to solve word problems because it involves the interaction of two symbol systems, and many of the students are weak in one or both of the systems. (Math teachers have always known that the biggest impediment to the learning of math for many students is language skills.) Indeed, the challenge of teaching word problems is so difficult that it needs the type of advanced approach made possible by constructivism.
In order to support a constructivist approach to learning how to solve word problems, Supermath created a new genre of software called Word Problem Processors, hereafter referred to as WPP. In WPP's students write stories to a lost and lonesome space creature whose disk ship has crash landed inside their computer, and who is stuck there and wants to be told a story. Writing stories is not a new math technique. What is new is that WPPs use artificial intelligence to enable the imaginary creature to parse students' language and to then determine if the language/story makes sense. If not, the creature gives students feedback about what the problem is with their story. Students then have to use the clue to revise their language before they can get any reaction from the creature. When a student makes a story that the creature understands and that involves math, then the creature produces a step-by-step solution. The role of the teacher is to come along and look at students' portfolios of stories and solutions and act befuddled and ask: "Why did the creature react mathematically to this story and words this way, and react in this different way to your other story?" This metacognitive prompt causes students to reflect about the creature's behavior and responses. The role of the teacher remains critical as the prompts are essential to channel students’ exploration into systematic learning. However, this process converts the teacher from a rule-giver and referee to coach, and converts the students from story problem solvers to individuals engaging in a sociological inquiry into the behavior of the space creature. It frees both from roles that previously both dreaded (if truth be told), and turns both into explorers.
The following is an illustration of the process for the simplest level of WPP, One-step Story Problems, though the process is the same for all the levels. In this level students create a story with three sentences called: Open, Change, and Question. Students easily construct each sentence by choosing from a menu of choices for sentence fragments. If, for example, students made the following story:
The creature then provides the step-by-step solution and answer, in this case the answer is obviously “5”. Students can then store their valid stories and solutions. At the higher levels the mathematics is far more complex with many steps in the solution. Yet, programming English is so complex that even at this simple, first, level, the software has over 1,000 grammatical rules built in.
This is a non-threatening process for students as there is no pressure for them to come up with solutions to the stories, i.e., problems. Instead, teachers focus on getting students to explain why the creature reacted to, i.e., solved the story, the way he/she/it did. (Students can choose to get the creature’s reactions in Spanish.) With sufficient experience with the above process students begin to understand how the space creature thinks about language and the nature of its mathematical reactions. With 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 problem.
Students engage in WPP’s in teams. After five or six days, each team has a portfolio of saved stories and solutions. Then the teams 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 can construct in their heads how the creature would react to it, which becomes how they react to it.
In order to make things interesting for the team competition, students can put in distracters and can convert the structure of the language in order to fool the other teams. [iv] When they do so the teacher again acts befuddled and metacognitively asks: "Why does the creature think this story is the same even though the language is different?"
For example, to fool the other teams, the team could first add a distracter to the above story, and change it to: Magazines cost $1.25. Joan bought 3 magazines. Jose found 2 magazines. How many magazines did Joan & Jose obtain altogether?
Then to further disguise it, the team can change the language structure to: 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 thing to fool them. So students experience the use of language in problems as a type of Play-Dough 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 effect of their language on the creature’s mathematical reactions (solutions) they construct their own understanding of how language and math interact, and are now able to solve word problems intuitively. Indeed, when we test students 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. In addition, even disadvantaged students come to enjoy solving word problems since they now view it as a sociological inquiry into the behavior of the creature. In addition, it has been reported that when they take a math test the students tend to report that the word problems were the easiest problems and that the distracters were so lame that they could not fool anyone.
Thus, the WPP Supermath 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 an increasing proportion of state tests. When disadvantaged students with experience in solving WPP’s come out of a test, they say that the easiest part of the test was the word problems, and that the distracters on the test were ridiculously lame and easy to spot. More importantly, our research shows that with this approach students can generalize this skill to word problems with different language structures that they have not experienced in the WPP. In addition, there are seven different levels of WPP’s, ranging from simple one-step problems to equations.[v]
The power of the learning drama approach is a function of the extent to which students are intrigued by the fantasy challenges. Indeed, if the developer does a good job of understanding what interests students, you do not need fancy graphics to intrigue them. Those who think that super fancy graphics are the answer, or even essential, should look at the primitive graphics used in South Park, Simpsons, etc. The best graphics are the graphics of the mind, and those are stimulated by interest. This also means that using technology effectively to teach math requires not only an understanding of math, but also 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 found ways to tap into kids’ interests and views to develop novel, creative, and effective learning dramas to teach most of the pre-Algebra concepts that are otherwise difficult and non-intuitive to teach and learn.[vi] Whether the curriculum has students trying to figure out if the Martian cashiers at the local MarsDonalds are trying to rip them off as a means of discovering the principles of rounding numbers, or has them playing golf to learn about angle estimation and measurement, or has them designing and testing a car to learn systems analysisstudents who previously were uninterested in, and doing poorly in math, suddenly are not only learning skills, but are also thinking mathematicallyand thinking that math is cool. Simply stated, Supermath is a curriculum that presents adult ideas from a kids point of view in ways that impact students cognitively and emotionally.
The second departure from conventional discussions of reform is that Supermath was committed to an exploratory problem solving approach that would increase traditional skills far more than traditional methods. In the November 2003 issue of the Kappan there were several articles that showed an example of constructivist learning.[vii] These examples are valuable, instructive, and inspirational in their own right. However, they do not provide the sufficient variety of activities needed to go beyond providing students with an isolated experience to what is needed; a sufficiently large set of related constructivist experiences that can cause students (and teachers) to change their view of mathematicsand to then apply that view to the learning of a large set of concepts. Supermath does fill that need. By providing a comprehensive set of 22 creative supplemental units with a consistent development framework and detailed teachers guides, Supermath provides the needed intensity of experience to impact students cognitively in ways that change how they 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 techniques such as constructivism and skills teaching and learning. They become a seamless whole. Schools and districts can easily integrate the use of Supermath units and skills in conjunction with state standards and student needs. Once students complete a Supermath unit they then practice and automate the relevant skills in conventional materials. However, the students now require significantly less practice than otherwise to automate the skills, and the acquired knowledge is in a more malleable form that enables them to apply the learned skills to a wider range of problems and situations. Our research shows that this mixture produces much more growth than simply using traditional methods 100% of the time. In addition, research based on the extensive piloting has generated recommendations of how to balance its use with traditional materials for a wide variety of program applications, from remediation to enrichment with Gifted students, and from inclusion in regular instruction, to use in summer and after school programs. [viii]
However, the power of the indirect approach to using software also creates a dilemma for evaluating, and preparing to teach, the materials. While the curriculum and software are easy to use and understand, the unique interface of software and curriculum in the indirect approach makes it strange, at first, to understand how these components work together instructionally for a given unit. In a traditional technology unit you can tell what is going on instructionally by either looking at the software or the accompanying materials. Both usually explicitly present content concepts. However, in the indirect approach of Supermath, where the software is just presenting a fantasy context, and the curriculum is guiding the teacher on how to introduce the concepts around this context, the two components are interpendent and synergistic at all times. In other words, neither component makes sense on its own, and you cannot tell what is going on instructionally by looking at just one of the components.
For example, trying to evaluate a given piece of software by itself, or trying understand a Supermath unit by just looking at the software, is not possible. As opposed to traditional math software, where it is obvious what skills it is designed to develop, when you look at a Supermath program you only see a disassociated activity that looks like a game with little or no obvious value or method for learning or teaching math concepts. Reading the curriculum by itself will make no sense given the constant references to occurrences in the related software. It is only when teachers or evaluators put themselves in the place of the students, and let the activities unfold for themselves as they would for the students, that the power and creativity of the approach of that unit becomes evident. Prospective users need to go through a given curriculum unit lesson by lesson, first reading the discussion in the curriculum for a given lesson, and then engaging in the called for computer activity in that lesson. Only then should they move on to the next lesson and repeat the process. (It is even better if this is done in a team with one person playing the role of teacher and the other the role of student.) Then, and only then, will they understand, and appreciate how the ideas unfold for the students over the course of a given unit and the true purpose of the software.
While the unconventional process of evaluating, or learning to teach, a Supermath unit may seem strange at first, once teachers are trained to use a few units the approach becomes second nature. At that point, when teachers first come to understand how a given unit expects students to figure things out, my experience is that teachers of advantaged students are convinced of its value, while teachers of disadvantaged students are often initially skeptical that their students will/can think mathematically the way that the curriculum calls for. 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 accomplishif teachers agree to trust the curriculum and the extensive piloting it has undergone. If teachers trust the curriculum and implement it, they become amazed at what their students are capable of, and interested in, doing. Students, in turn, become amazed to discover that they can figure out math rules and enjoy math.
As a result, some initial staff development on high priority units is essential to learn how to integrate the curriculum’s lessons and software in the classroom, and most importantly, to trust the bold approach of the materials. However, once this is done, a school and/or district has developed a practical and inexpensive way to responsibly incorporate advanced teaching techniques in ways that both increase student learning and appreciation of mathematics.
Creating this bridge between mathematical reasoning and increased skill acquisition requires schools and dedicated teachers who are willing to take a chance on using a very different validated approach part of the time. The natural tendency of schools will be to meet the NCLB math accountability requirements by cramming as much test prep/content drill as possible. While this strategy will increase test scores for a time, the gains will quickly plateau, which is unacceptable under NCLB. In addition, relying on cramming for the test will also produce another generation of math haters and mathphobics.
Alternatively, Supermath represents a new curricular approach that combines the best of traditional and progressive approaches in a data driven balance that enhances both. There is no need for math “wars” (…or reading “wars”). It is indeed 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 is needed are new types of dialogues and more creative, powerful, and synergistic forms of curricula.
[i] John Marshall, “Math Wars: Taking Sides,” Phi Delta Kappan, November 2003, 193-200)
[ii] As an illustration of how divergent the Supermath environments were, one of the math educator reviewers for NSF became incensed at the golf application on the basis that it was not an “authentic” application since golfers really do not 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. She had not bothered to read the curriculum and therefore did not realize that the whole point was that it was designed so that the ball would not go in the hole unless you used decimal length since a major goal was to reinforce the use of decimals. 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.
[iii] Later on when students are making more complex multi-step stories, if they make a story that involves only one calculation, the creature gets insulted again because it now wants to do several calculations. The sophistication of the creature’s expectations evolves as the level of the problems increases.
[iv] It is important for students to generalize solutions to different forms of language since research has generally shown that it is possible to get students to solve a particular form of word problem, but they become confused when the language changes, even if it is the same problem.
[v] 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.
[vi] For a complete listing of Supermath units and skills, contact the author.
[vii] See: Thomas C. O’Brien and Judy Barnett, “Fasten Your Seat Belts,” Phi Delta Kappan, November 2003, 201-206, and Carole Funk, “James Otto and the Pi Man,” Phi Delta Kappan, November 2003, 212-214.
[viii] For example, a good combination for improving test scores seems to be about 40-50% Supermath and 50-60% traditional for a period of time for disadvantaged and under-performing students, while a lower percentage of Supermath activities are needed for advantaged students.