Warning: unpolished thoughts (a.k.a. rambling) follow...
A colleague recently sent an article describing the age-old problem of math topic coverage being a mile wide and in inch deep. A few excerpts from the article are enough to illustrate the main points math curriculum crisis transpiring in the state of Florida right now:
"The problem with the old standards, educators say, is that many topics were covered lightly and then repeated year after year."and
"Because the state's annual test measured performance on the 89 standards, Kuches and other teachers had no choice but to race through them in the hope that students would learn. The results were not always good, especially for slower learners."I used to think that this problem could be quickly fixed...just teach fewer topics and explore them from many different angles so that ALL students can understand them. After a dozen or so semesters teaching, Geometry, I still think this is a good idea in theory, but I have also realized that some young minds just aren't mature enough to comprehend certain math topics. Think about Bloom's Taxonomy (or a more specific math example, van Hiele levels of geometric reasoning) and how some students struggle with the knowledge or comprehension level of many topics in school. In Geometry, our curriculum (i.e. proofs, inductive and deductive reasoning) implies students should be synthesizing and evaluating geometric ideas. From my experience, that's just not cognitively possible for some students as fifteen or sixteen year-olds. It might not be possible until they are seventeen or eighteen. If the topics don't repeat themselves, at least a little bit, students will be missing out on the opportunity to be exposed to these ideas later on in their "math career."
I do like the approach of only teaching several big ideas each year during middle school math, but I'm just not sure if its reproducible at the high school level. Some spiraling of topics seems to make sense, but how much is too much given what we know about adolescents' cognitive abilities? I am by no means an expert in adolescent psychology, so I'm hoping y'all are able to chime in with your thoughts and experiences.