Depth versus breadth quandry

Depth vs. breadth in the American curriculum. Who understands it? The Third International Math and Science Study and the Midcontinent Regional Educational Laboratory have characterized the nation's curriculum as a mile wide and an inch deep. The McREL report studied the standards adopted by many states and suggested it would take an additional nine years of education to teach it.

Here in Nevada, the Legislature heard so many concerns about the state's new academic standards they passed Senate Bill 466 which requires the Council to Establish Academic Standards to review, revise and prioritize the standards they just adopted last year. In Nevada, like most of the nation, approximately 50 percent of students are failing the math section of graduation exams. Those exams look much like the current test in Nevada. Our new academic standards which will be tested in 2001 look like Arizona's, only 11 percent of the students passed their new test. I hope the state has lots of money available for remediation efforts.

The biggest problem the council will face is whether they will revise standards so they are both appropriate and reasonably attainable for all students or will they be more concerned with how their actions may look?

The decision they make will determine if the state will require mastery of the material taught in school or just plain coverage. Both the TIMSS and McREL suggest that if too much material is being introduced and taught, students cannot master the material.

That lack of mastery results in students learning material for a particular test but not understanding how it might be connected to other things they have learned or practical applications. Then, because of the scope of the academic standards, there is very little or no time to regularly review important information so students are more apt to forget. Many of the students that I teach at the university have no idea that the Set Theory they learned was in any way connected to Venn Diagrams. Too many students saw those as two different disconnected concepts. When those two mathematical ideas were connected to logic, students seemed almost stunned. But the fact is some of those students will go on to take a course in Boolean Algebra and realize if they understood sets, Venn Diagrams, or logic, then they would also understand Boolean Algebra. They are the same. Far too few students will ever see the connection between probability and what they learned in Set Theory or Venn Diagrams.

The biggest difference is in Set Theory, we talk about unions, intersections and complements. In Venn Diagrams we express those ideas in pictures. In logic, the same subject is discussed in terms of conjunctions, disjunctions and negations. If the standards were not so dense, then there would be time to connect these concepts. Those connections would review and reinforce previously learned material, resulting in greater student understanding and application.

When the math standards were originally adopted by the council, process standards were included to ensure students were shown these connections. However, after the council approved those standards, they allowed others to move material around grade levels and add more material. This, in essence, rendered the process standards irrelevant because of a lack of time. That action clearly demonstrates to me that neither the council members nor their high paid Washington based consultant understand the "breadth vs. depth" concept. Their actions would seem to indicate that more means more rigorous. Actually, it more means less rigor.

Let's look at percents. Most of us can find the percent correct on a test if they had 18 right out of 20. Most could find the discount on an item if the price was $68 and went on sale at 20 percent off. However, students that don't understand ratio and proportion might run into difficulty if they were asked to find the original cost of an item that was marked down 20 percent and purchased for $68.

If that last problem would cause difficulty, then what would happen if I asked students to determine how much a dress should be priced if it was bought wholesale for $300 and the retailer wanted to put it on sale at 20 percent off and make a 25 percent profit. The answers to above problems are 90 percent, $13.60, $85 and $468.75. Don't assume you can do these. Work them out and see if you get the correct answer.

Heck, who knows, with time when students were taught percents, they might also learn about simple and compound interest. Wouldn't it be nice if students could explain the relationship between fractions, decimals and percents - not just recite some equivalences.

If we expect kids to know this, we better give their teachers time to develop the concepts and make the appropriate connections so students are more comfortable in their knowledge and understanding that will allow them to apply it.

If those questions are appropriate for all students to know upon graduation, the Council to Establish Academic Standards has to determine if the material is reasonably attainable. In other words, how much time are they going to allocate to learning this and how much time will be built in for regular reviews to reinforce that knowledge?

Bill Hanlon, a Las Vegas educator, is a member of the Nevada Board of Education. His views do not necessarily reflect those of other members. His e-mail address is


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