3/29/98
     Colorado's math standards deserve nothing more than the dismal D grade they received in the Fordham Foundation's recent nationwide evaluation. They're mush. 
     In Colorado's standards ``the text is often vague, indefinite or careless,'' say Fordham's reviewers, Ralph A. Raimi and Lawrence S. Braden. If that sounds harsh, it's because they are applying mathematicians' standards of precision. Raimi is professor emeritus of mathematics at the University of Rochester (New York) and Braden, who teaches at St. Paul's School in New Hampshire, is a past winner of the Presidential Award for Excellence in Science and Mathematics Teaching. 
     So blather like ``recognizing when a pattern exists and using that information to solve a problem'' and ``students link concepts and procedures as they develop and use computational techniques'' doesn't impress them. 
     Colorado is not uniquely bad; in fact, half the states were worse. 
     Summing up, Raimi and Braden write, ``The principal failures stem from the mathematical ignorance of the writers of these standards, sometimes compounded by carelessness and sometimes by a faulty educational ideology.'' 
     Many of the tenets of that ideology are drawn from a single source, the so-called ``standards'' issued by the National Council of Teachers of Matheamtics in 1989. The NCTM adopted the view that children must rediscover for themselves the principles and the techniques discovered over centuries by some of the most brilliant and creative people who ever lived, and that children can best do this by solving problems somehow related to their lives. 
     At best this is inefficient, and at worst it is positively harmful. Many children who are perfectly capable of learning how to do, say, long division, fail at the far more difficult task of reinventing their own algorithm and come to rely almost totally on the calculators they are encouraged to use from kindergarten on. 
     The glossary for Colorado's standards does not mention long division, though it gives a definition for ``basic facts.'' 
     The state math standards open at the highest levels of abstraction with ``six goals'' for students: become mathematical problem-solvers, learn to communicate and reason mathematically, make mathematical connections, become confident of their abilities and learn the value of mathematics. (All the standards are on the state Department of Education's Web site, www.cde.state.co.us.) 
     No one could object to these, but neither could anyone use them to decide what should be taught to children in each grade. 
     At the next level, six standards cite number sense, algebraic methods, data collection, geometric concepts, measurement and computational techniques, but all with a common refrain: students will use mathematics ``in problem solving situations and communicate the reasoning used in solving these problems.'' 
     That fundamentally misconceives the nature of mathematics, which is a deductive system and not an empirical science. Of course mathematics is useful in practical contexts _ miraculously successful, Raimi and Braden note _ but children are impoverished if they never learn that it is also an intellectual adventure. The Colorado standards never mention axiom, theorem or proof. Instead, children are supposed to reason ``informally'' about geometric properties. 
     For each of the six standards, there are _ finally _ a few sentences describing expectations for different grades, clustered as K-4, 5-8 and 9-12. But they are still a long way short of specific. ``Using number sense to estimate and justify the reasonableness of solutions to problems'' occurs at all three levels, first with whole numbers, then rational numbers and finally real numbers. 
     If you collected all 28 sentences covering K-4 math, you still wouldn't know what to teach when. 
     In useful contrast, E. D. Hirsch's Core Knowledge math curriculum has four to six pages for each grade listing the topics that will be covered and what children will ``know and be able to do.'' 
     The bitterest irony for Colorado is that its standard-makers deliberately modeled their work on California's 1992 Mathematics Framework. But as California's test scores fell into the national cellar, a coalition of outraged parents and professional mathematicians rejected the framework, and prepared a new set of math standards that the California Board of Education adopted in December. It was clear, explicit and rigorous. 
     California got an A.