Dipping math scores heat up debate over math teaching psychologists differ over the merits of teaching children ?whole math.?

By Bridget Murray
Monitor staff

Biting their lips in concentration, seventh-graders in John Phillips? pre-algebra class squint at the problem he has chalked onto the blackboard: -11 + (+8) = ?

Nobody is scribbling down a formula, however. Instead, the students shuffle red and yellow plastic disks across their desks. Each student plunks down 11 red disks for -11 and 8 yellow disks for +8. When they match them disc for disc, three reds are left.

'It?s negative three,' the students call out.

In another classroom here at Gunston Middle School in Arlington, Va., Linda Allen teaches seventh-graders proportions and percentages using their mothers? ages. The average mom?s age is 30, it turns out, so Allen throws in her anomalous 60-year-old mother and shows how an extreme number can skew the data.

Allen and Phillips use a demonstrative approach to mathematics teaching that has gained popularity during the 1990s. Often referred to as 'constructivist math,' the approach seeks to enliven a potentially dry subject. Also called 'whole math,' 'situated math' and 'authentic math,' the approach connects children's informal math understanding?of objects and people?s ages, for example?with abstract math concepts.

'Instead of just feeding kids the formula for a triangle, you show them how to get there,' says Phillips. 'Showing them the ?why? of math helps them apply and relate math concepts.'

Yet for all its popularity, the approach also has its detractors. Some psychologists and educators claim the method undermines children?s grasp of math fundamentals?the formulas and numerical operations that constitute math?s core. They say children need those solid math skills before they can truly connect math concepts.

Recent findings from international math testing have heightened the controversy. Released in February, results from the Third International Mathematics and Science Study (TIMSS) peg American high school seniors third from last in mathematics literacy out of 21 nations. The U.S. Department of Education calls such innumeracy 'unacceptable,' in an economy built on computers, currency and data. The world?s top jobs increasingly demand higher math, U.S. education officials say.

While some educators blame the poor results on schools? move toward constructivist math, constructivists claim that schools? continued use of skill-and-drill methods is the real culprit. Meanwhile, a vanguard of psychologists and educators believe that the best math teaching blends both approaches.

Seeing the numbers

Constructivist math gained momentum in 1989 when the National Council of Teachers of Mathematics endorsed a new set of standards. The standards urge teachers to relate problems to 'everyday mathematical situations' and to use teaching tools such as cubes, grid paper and computers.

The approach draws on research suggesting that children actively construct knowledge, rather than passively absorbing it from teachers. Constructivist teachers tie children?s intuitive feel for math?gleaned from experiences like telling the time and playing board games?with classroom math, says Herbert Ginsburg, PhD, a psychologist and mathematics researcher at Columbia University?s Teachers College.

'Instead of segmenting math knowledge by lesson or textbook chapter, you integrate new concepts with what children already know,' says Ginsburg, who explores ways to gauge children?s math development in his book, 'Entering the Child?s Mind: The Clinical Interview in Psychological Research and Practice' (Cambridge University Press, 1997).

A constructivist approach encourages students to test their own math theories through group work and discussion, says psychologist James Greeno, PhD, an education professor at Stanford University. His program, the Middle School Mathematics Through Applications Project (MMAP), helps students do that. The program challenges small groups of students with math-based computer simulations. In one simulation, students design offices for an Antarctica-based research team. In drawing up the blueprints, they calculate the floor measurements on a dot grid. In another simulation, students analyze population fluctuations among Alaskan caribou and wolves.

The simulations make math memorable by demonstrating its usefulness, says Greeno, who published an article about MMAP in APA?s American Psychologist (January 1998, Vol. 53, No. 1, p. 5?26). Hands-on tools, or manipulatives, also help elucidate math, says Phillips at Gunston Middle School. Counters demonstrate positive and negative integers, and colored blocks illustrate the area of a square, he says. The tools especially benefit students who need extra help, he says. As one of his students puts it: 'They help you see and touch the numbers.'

Back to front

Despite the positive reaction to constructivist math, educators aren?t sure how much U.S. math classes actually use it. The approach?s supporters claim that U.S. students? low TIMSS scores suggest it isn?t used enough. Its critics claim just the opposite. They say the TIMSS scores cast doubt on constructivism?s effectiveness, a position that may be prompting a pendulum swing back to basic math. Responding to the U.S. Department of Education?s call for tougher math in middle school, many states are adopting rigorous standardized tests, more schools are pressuring students to take algebra in eighth-grade and the bellwether state of California recently adopted stricter math standards.

Critics of constructivist math welcome this shift. One of them is psychologist David Geary, PhD, who says constructivist math conflicts with cognitive findings that math learning is hierarchical.

'The constructivist approach is back to front,' says Geary, a cognitive researcher at the University of Missouri. 'If you don?t have the pieces first, you won?t understand the whole. You can?t do algebra without multiplying and dividing first, and you can?t multiply and divide without knowing addition and subtraction.'

Math learning is not automatic like language acquisition, says Geary. Students need directed instruction because they aren?t naturally motivated to learn, as the constructivists believe, he says. In a recent study of 372 Chinese and American sixth-graders, 12th-graders and older adults, Geary found that Chinese children outperformed American children on computational and reasoning measures.

However, Chinese and American older adults performed much the same, and IQ scores in the two countries were also similar. The study appeared last year in the Psychonomic Bulletin & Review (Vol. 4, No. 3, p. 425?430). According to Geary, America?s math performance sags because our country lacks a rigorous, sequenced national math curriculum like China?s.

'If you?re skipping drill and practice, you?re just skipping the hard part,' he says.

Psychologist Robert Siegler, PhD, a math researcher at Carnegie Mellon University, agrees.

'If schools don?t teach low-level skills to a high degree of mastery, students won?t be able to solve more demanding problems,' he says. 'You can?t do the word problem until you first memorize the multiplication tables.'

Linda Allen of Gunston also observes that students don?t always connect hands-on approaches with abstract operations. Some students find math tools distracting, or they can?t extract the math from the context, she says.

A blend of both?

Many?perhaps most?psychologists value math drills and constructivist math and think that schools should teach both, not one or the other.

Psychologist Daniel Berch, PhD, of the National Institute of Child Health and Human Development, believes the debate has grown overly politicized and acrimonious. American math education needs a middle ground, he says.

'If we want to improve math teaching, we need to look at the research,' says Berch. 'Instead we?re getting into a false debate just like the one over whole language and phonics.'

If the two math camps actually sat down and talked?as the whole language and phonics people recently did?they?d find considerable overlap in their approaches, says Thomas Carpenter, PhD, an education professor at the University of Wisconsin and co-author of 'Making Sense: Teaching and Learning Mathematics With Understanding' (Heinemann, 1997). Both sides have exaggerated and demonized each other?s stances, without noting where their approaches intersect, he says.

And, in fact, many teachers concoct their own blend of constructivism and basic math. Allen and Phillips illustrate their lessons with hands-on tools and real-world examples, but they also assign homework problems and test students regularly.

'It?s so easy in education to get swept up in one way or the other, but there?s never a ?the way? to do anything,' says Phillips. 'What you need is lots of different styles for lots of different kids.'

Further reading
For more information on math education, consult these books:
? 'Why Numbers Count: Quantitative Literacy for Tomorrow?s America,' by Lynn Steen (College Board, 1997).
? 'Children?s Mathematical Development: Research and Practical Applications,' by David Geary (American Psychological Association, 1994).
? 'The Challenge in Mathematics and Science Education: Psychology?s Response,' by Louise Penner et al., (American Psychological Association, 1993).