Barry Garelick: Trespassers in Wonderland

(This piece originally appeared in Kitchen Table Math, October, 2005: .  It is just as relevant today as it was then.  Reprinted by permission)

by Barry Garelick

Only 20 percent of fourth graders correctly calculated the answer to 314 x 12 on the 2004 National Assessment of Education Progress (NAEP) exam. Eighth graders’ performance was also disturbing: a question asked for the length of a line segment above a ruler, with one end at the 2 cm mark and the other at the 7 cm mark. Only 58 percent of eight graders got it right; and it was multiple choice. On the international front, anyone following how U.S. fourth and eighth graders fare in international tests in math (called TIMMS) have by now noticed that U.S. has come in about 14th or 15th, and that Asian countries top the list (Singapore is number one).

To put the issue of math education in context, one has to understand the prevailing attitude toward math education in this country. Two years ago, at a packed conference on math education, Jim Milgram–a math professor from Stanford (now retired)–presented the following story problem which, he noted, seventh grade students in Russia are expected to solve:“Two people left their villages at sunrise and walked, each to the other’s village at constant speed. They met at noon and the first arrived in the others’ village at 4:00 PM while the second arrive at 9:00 PM. What time was sunrise?”

At this, a man sitting behind me huffed “Who cares?” This is a fairly typical reaction. Many people believe that U.S. students do not perform well in math because they are not taught how to apply it to real-world, relevant problems.  That the problem is challenging was of no concern to the commenter.

Sentence first—verdict afterward

In May, 2005, the National Council of Mathematics Teachers (NCTM) in a statement which appeared in the Washington Post echoed the exact same “Who cares?” sentiments as the disgruntled man at the conference:

“For generations, mathematics was taught as an isolated topic with its own categories of word problems. It didn’t work. Adults groan when they hear ‘If a train leaves Boston at 2 o’clock traveling at 80 mph, and at the same time a train leaves New York…’ Whatever problems and contexts are used, they need to engage students and be relevant to today’s demanding and rapidly changing world.”

NCTM is a large organization based in Reston, Virginia which exerts considerable influence over how math is taught in this country. In 1989, NCTM published a set of curriculum and evaluation standards for math, and revised them in 2000. Some states have relied on these standards in framing their own. Such standards de-emphasize learning basic skills, is light on content and heavy on context-based learning otherwise known as “real life math”. Cathy Seeley, current president of NCTM is critical of math texts and programs that tell students “here’s the rule, now do the problem” and says there is too much “teacher instruction” in the U.S. NCTM’s topsy-turvy approach to teaching math is more like “Here’s the problem, you figure out the rules needed to solve it”—an approach alarmingly similar to the Queen’s declaration at Alice’s trial in Alice in Wonderland: “Sentence first–verdict afterward.”Some real life problems

Here’s an example of a real life problem which can be found on NCTM’s very own web site in the section called “Illuminations”:

“Suppose you have saved $63. You find a used video game system that you would like to buy. The seller is asking $180. You earn $10 a week doing odd jobs. How long will it take you to earn enough money to buy the game?”

While this type of problem has been around for years, NCTM’s suggestions for how to “explore” the problem in class is what’s different. They explain that adults typically subtract 63 from 180 and divide by 10. While this would be a preferred approach for students to have mastered by the 5th or 6th grade–the grade level for this activity–NCTM describes with particular pride a student entering 63 into the calculator (no apology was offered for calculators being used here), then adding his first week’s allowance, then the second, third, and so forth until the display showed that he had at least $180 (12 weeks).

NCTM explains: “Allowing students the freedom to use strategies that are intuitively obvious to them helps them to feel more comfortable in the problem-solving process. At some stage it also helps them appreciate the efficiency of standard algorithms.” NCTM does not discuss when this stage will occur. One would hope that it occurs quickly so that the calculator-aided counting-on-fingers method can be supplanted with the more efficient method that students in Japan and Singapore have mastered by the third grade.

A Word from NCTM

Recently, when asked why U.S. students suffer from an inability to perform complex reasoning and mathematical assignments compared to students overseas, Cathy Seeley (then the president of NCTM) responded: “We’re not doing as much problem-solving of that type as we need to be.” In another instance, she said “We can definitely learn lessons from Singapore, Japan and China. But we have to look beyond their textbooks to determine what these lessons are.”

Even a faithful NCTM adherent would not fail to notice that in Singapore’s textbooks, problems require multi-step solutions that are considerably more complex than what we expect US students to solve at that grade level. From a sixth grade Singapore textbook: 3/5 of Mary’s flowers were roses and the rest were orchids. After giving away ½ of the roses and 1/4 of the orchids, she had 54 flowers left. How many flowers did she have at first?

Looking beyond the textbook as Cathy suggests allows NCTM to throw the baby out with the bath water, and to reject problems that are good by saying “It’s not the text, it’s the teaching.” In fact, in Japan, Singapore and Russia, they do teach math differently. They teach it correctly. They teach content. They teach skills and facts as a foundation upon which understanding will be built. They teach like they used to in the U.S.

Alice’s retort

In Alice in Wonderland, Alice tells the royal family “Who cares for you? You’re nothing but a pack of cards!” There are many packs of cards at work in education. It starts with education schools that propagate the philosophy that knowledge must be top down, rather than “skills-based”. Boards of education, school districts, departments of education and of course NCTM follow the ed school lead and have become packs of cards. The result is that math education is almost content free. Anyone who disagrees with such philosophy is wrong and told “Off with your head”. Over and over, as parents, teachers, and world-class mathematicians protest how math is being taught, and tell school boards and administrators the type of content students should be mastering, they are viewed as trespassers in Wonderland. Story problems are met with groans, proclaimed not to be real life, and dismissed with a mighty “Who cares?”

“Who cares is not the point,” Jim Milgram says. “Let me give you an example of a problem that people had better care about since it will affect their very lives. Design a robot arm to select and lift items off an assembly line and place them on a second line correctly positioned for a second robot to work on them. There is no chance in hell that someone can do this if they can’t do the Russian problem about the two villagers.”

Until reform math is recognized for the pack of cards it is, the influence of NCTM and their followers will continue. The wake of this influence engulfs our children, many of whom do not know how to multiply two-digit numbers without a calculator, nor how to use a ruler.