Reform Mathematics Education
How to “Succeed” Without Really Trying
Cofounder, Mathematically Correct
Since the 1980’s, there have been substantial efforts nation wide to weaken mathematics education in America, and these efforts have largely been successful. This is not a communist conspiracy [Note 1]. It flows from an honest desire to help the less fortunate. This effort is based on the misguided notion that weaker mathematics will be helpful to the traditionally disadvantaged groups in our society. It is this effort, curiously known as reform, that is the root cause of what has come to be known as the math wars.
You won’t find many reformers who will openly admit that they favor “dumbed-down” mathematics. In fact, the reform movement is characterized by a plethora of rhetoric to the contrary. The diatribes are extensive and frequent and are laden with phrases like “higher order thinking” and “conceptual understanding” and “real-world problems” while shy on terms like “arithmetic” and “algebra.” Reformers have learned their scripts well, and the rhetoric comes gushing forth with little provocation.
The conditions that prompted this movement are obvious. Poor people, minorities, and women are under-represented among those who reach high levels of mathematical achievement. Those who cannot master arithmetic and algebra are unlikely to achieve a decent college education. There is no question that the educational system in this country is not successful for a great many students.
One way to deal with this problem is to make the mathematics easier. This means less rigor, less emphasis on arithmetic and algebra, more reading and art and creative projects, less emphasis on correct answers, more calculators, and a host of other reform-minded solutions. Stylish pedagogical methods combined with rhetoric about higher order thinking while downplaying or condemning outright both computation skills and mathematical proof complete the package. This is reform mathematics education.
Sometimes dubbed traditional or anti-reform, the second perspective has come in abreaction to the first and is mainly supported by parents and mathematicians. This perspective holds out that the mathematics must not be “dumbed-down.” The key in this perspective is to increase achievement rather than to decrease expectations. Central to this position is that the traditionally less fortunate are not well-served by weaker mathematics and, in fact, should be insulted by it. The real key to success is real mathematics achievement, and every effort should be made to foster this achievement.
Ironically, the struggle to promote real mathematics education is left up to those outside of the field – mostly parents. The perspective is traditional in the sense that it seeks to prevent learning expectations from being further eroded away by putative reform efforts. Mathematics education in America has not been very successful. However, do not look for relief in the reform notions. We would be better off if all the energy behind the reform was redirected toward clearly defined achievement goals and we measured progress toward those goals frequently and objectively.
Obliterating Distinctions between Success and Failure
The reform designs open the door to claims of successfully teaching mathematics without really doing so. The reform writings and methods are many and varied, but a common feature is that they end up obscuring the failure to teach mathematics. In reform mathematics education, the goal of success for all is not supported by achievement but rather by redefining success and, mostly, by obscuring failure. Here are but a few examples:
|Group Learning and Group Tests – The story of Apollo 13 is used to promote group learning and group assessments with the argument that our students must learn to work together like people do in the real world. Never mind that people in the real world don’t sit in groups doing algebra problems. Group learning is plagued by inequities that most parents identify quickly – some do the work while others learn that they can “succeed” without learning the material and without effort. Group assessments effectively erase the ability to monitor individual achievement or to provide useful diagnostic information. Whether or not individuals are learning is obscured by these methods.
|Calculators – Many argue that routine skills are out of date, and that technology has changed the mathematics that today’s students need to know. The position includes multiplication and division, obviously. However, today’s calculators can manipulate fractions and solve equations as well. Distancing students from these activities takes away the learning experiences that help form the foundation of mathematical understanding. By far, most American parents want their children to be able to solve problems without calculators. The reliance on calculators allows reformers to claim success even when children do not learn the fundamental operations of arithmetic. Soon they will claim success in algebra for students who have not learned how to solve equations.
Authentic Assessment – One of the greatest evils from the reform perspective is objective testing. It would have to be because these measures can identify failure. Many arguments are advanced for this perspective, but addressing them in detail is beyond the scope of this report [Note 2]. The proposed alternative is frequently called authentic assessment. Translating this bit of jargon into English isn’t easy. Basically, it refers to a variety of procedures that involve less mathematics, more writing or talking, and very subjective evaluation. In the worst instances, students suffer if they do not support the intended politically correct perspective in their response. But, politics aside, these methods are reliably unreliable. The subjective nature leaves little opportunity for valid information to be obtained. Sometimes, one cannot even tell who actually did the work. In the long run, many invalid assessments tend to average out (false equity) and, again, real differences in achievement go undetected.
The first stage of this process, examining the content of the school program, can be a little easier for parents who make use of existing resources that identify content by grade level. Coming on the heels of failed reform efforts in California, expectations for achievement that are roughly in line with those of the most successful countries of the world were developed. These documents identify achievement levels in terms that are sufficiently clear for parents to evaluate. Parents are encouraged to measure the school programs against these contents as a way of finding out whether or not important content is being covered.
With the aid of these materials, parents can more easily find what is present and what is absent in the programs used in local schools. These documents enable parents to match local content to grade levels according to high-level standards.
|Projects – The reform programs are loaded with projects and activities, often called investigations. Part of the argument for these methods relates to stimulating student interest. There are also claims of richer mathematics and the importance of context. Even a casual inspection of these activities will show that they tend to be very time consuming while involving very little mathematics. Time for mathematics, both in class and at home, is seriously limited and must be used as efficiently as possible. These activities are inefficient learning methods. But, beyond that limitation, they promote the evaluation of students on the basis of non-mathematical dimensions such as how artistic the display is or the writing style of the report or the social value of the application.
|Standards – The reform movement claims to be based on standards, although most parents will be surprised by what they find – and what they don’t find – in reform standards documents. It is contrary to the goal of the reform to produce explicit statements about what students know and should be able to do – again, spotting failure would be too easy. Consequently, the reform movement produces standards that are so vague that one cannot tell whether they have been met or not. Any attempt to write tests for these standards, for example, will be unreliable because the required content is unclear. Reformers hate lists of clearly stated objectives and call them laundry lists. However, vague learning expectations are effectively the same as no learning expectations at all. Again, it becomes impossible to differentiate success from failure.
|Strands – When attempts are made to subdivide mathematics into content areas, such as algebra and geometry, the subdivisions are often called strands. The reform movement uses this technique while simultaneously avoiding explicit content. Thus, all of the elementary school work with arithmetic falls into one strand which becomes just one of many topic areas students are supposed to address. The consequence is that students can still succeed while failing in arithmetic. The same thinking reduces algebra to just one component of mathematics in later grades with similar consequences.
|Pedagogical Fads – The reform movement places great emphasis on classroom methods, such as those that involve groups, calculators, activities and projects, manipulatives, explorations, art work, and non-mathematical themes. Irrespective of any relationship between these methods and learning (or lack thereof), there are consequences of the fact that the emphasis on these styles is pervasive in reform documents. Even reformers bemoan the fact their followers often carry out reform by adding a few new gimmicks to their bag of classroom tricks. The heavy emphasis on style quite naturally takes attention away from mathematical content. As teachers attend to implementing these processes, their evaluations of students become biased toward process and away from content. Mathematical learning will often take a back seat to artistic ability, cooperation, or even political correctness again blurring the distinctions between success and failure when it comes to learning mathematics.
With the demise of our ability to differentiate success from failure, the reform movement will claim broad successes. School systems in America have the uncanny ability to claim improvements and reforms year after year while the content is gradually leeched out of the system. Meanwhile, fewer students will suffer wounds to their self-esteem because their failures will go undetected. Such a system will identify fewer failures among poor and minority group students, so reformers will claim a victory for equity.
Unfortunately, success in this approach will have lost its value. The claims of success operate like social promotion as applied to education bureaucrats. We may gain some “equity” at the cost of achievement, but the more advantaged parents will continue to find ways to make sure that their children learn in spite the best efforts of the reform-minded. Meanwhile, the net effect of the reform will be further deterioration in the mathematical abilities of America’s youth. The majority of these students will not find alternative forms of education to make up this deficit. It is from this majority that we will draw our next generation of teachers.
Note 1: Although not a communist conspiracy, there is some justification for the belief that some sort of conspiracy is at work. The reform designs are heavily promoted by the National Council of Teachers of Mathematics (NCTM). In turn, the educational branch of the National Science Foundation (NSF) then funds the development of curriculum materials that align with the NCTM dictates. The products of these efforts are then advertised by the U. S. Department of Education, while the NSF pushes for their adoption by states and districts.
Note 2: The interested reader should see Chapter 6, “Test Evasion,” in The Schools We Need and Why We Don’t Have Them by E. D. Hirsch, Jr., Doubleday, New York, 1996.