focus, rigor, coherence

What's Missing from Math Standards?
by William H. Schmidt
American Educator, Spring 2008

Why do some countries, like Singapore, Korea, and the Czech Republic, do so much better than the United States in math? I've heard all sorts of reasons; diversity and poverty top the list. But after some 15 years conducting international research, I am convinced that it's the diversity and poverty of U.S. math standards—not the diversity and poverty of U.S. students—that are to blame.

The single most important result of the Third International Mathematics and Science Study (TIMSS) is that we now know that student performance is directly related to the nature of the curricular expectations. I do not mean the instructional practices. I mean the nature of what it is that children are to learn within schools. (In the U.S., the curricular expectations are usually referred to as standards; in other countries they are known by various names.) ...

The TIMSS research has revealed that there are three aspects of math expectations, or standards, that are really important: focus, rigor, and coherence. ...

Focus is the most straightforward. Standards need to focus on a small enough number of topics so that teachers can spend months, not days, on them. I'll just give you one illustration: in the early grades, top-achieving countries usually cover about four to six topics related to basic numeracy, measurement, and arithmetic operations. That's all. In contrast, in the U.S., state and district standards, as well as textbooks, often cram 20 topics into the first and second grades. That's much more than any child could possibly absorb.

Rigor is also pretty straightforward—and we don't have enough of it. For example, in the middle grades, the rest of the world is teaching algebra and geometry. The U.S. is still, for most children, teaching arithmetic. It's not rocket science: other countries outperform us in the middle and upper grades because their curricular expectations are so much more demanding, so much more rigorous.

Coherence is not quite as easy to grasp, but I believe it is the most important element. Coherent standards follow the structure of the discipline being taught. All school subject matter derives from some academic discipline, be it geography, history, mathematics, physics, etc. Once that formal academic body of knowledge has been parsed out and sequenced from kindergarten through 12th grade, it should reflect the internal logic of the discipline. This is especially important in mathematics, which is very hierarchical. Topics in math really need to flow in a certain logical sequence in order to have coherent instruction. If you look at the math curricula of top-achieving countries, you see a very logical sequence (which I describe in the sidebar on the right). The more advanced topics are not covered in the early grades. Now, that seems obvious—until you look at state and district standards in the U.S. Everything is covered everywhere. Far from coherent, typical math standards in the U.S. often appear arbitrary, like a laundry list of topics.

Thematic teaching cannot reflect the internal logic of the liberal arts disciplines.

Histogeomegraph is not a discipine.

Histogeomegraph is a mash-up.