A Quest for Understanding
Reprinted from "Parent News"
February, 2005
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Mark Sawula, Blake’s first Pre-K–12 department chair in math, provided the following article.
In the 16th century, the Italian Jesuit Matteo Ricci journeyed to China to teach techniques for ordering and remembering vast quantities of information. Ricci counseled his students to construct “memory palaces,” many-chambered mental edifices that housed scores of mnemonic icons. In Europe, Ricci’s principle of relying on the power of spatial memory was hardly new — more than 1,500 years earlier Cicero described a version of the same technique. For hundreds of years, scholars held memory skills in high regard. The memory arts faded into obscurity only after the influence of Gutenberg’s revolutionary printing press became so pervasive that plentiful published materials obviated their need. In a manner of speaking, technology changed the curriculum.
Today we live in the midst of another momentous technological revolution. The influence of information technologies, particularly those related to computation and telecommunication, pervades our lives. Our businesses, governments and schools are awash in data. The human genetic code itself is being subsumed in the digital database of knowledge. And even mathematics education feels the effects. While data analysis has moved into a position of unprecedented prominence, scientific calculators have pushed the algorithm for finding square roots right out of the doors of our high schools. Now, relatively inexpensive computer-algebra systems found on the most advanced graphing calculators can manipulate and solve any equation or system of equations in the high school curriculum. With such powerful tools increasingly available, what elements of the math curriculum will be rendered obsolete?
Another major pressure on the math curriculum is also born of the Information Age: international comparative studies. In study after study of eighth graders (see www.pisa.oecd.org and timss.bc.edu), the United States underperforms most other industrialized nations, particularly Belgium, Hong Kong-China, Japan, Korea, the Netherlands and Singapore. In the Programme for International Student Assessment 2003 report, even the top 25 percent of our students suffer in comparison. Analyses suggest that in higher-performing countries, teachers cover fewer topics and skills but develop deeper conceptual understanding for the topics they do teach. In the United States, there is an undue emphasis on algorithmic plug-in of procedures at the expense of genuine reasoning. Other studies show instruction that initially focuses on procedural skills tends to hinder the development of deeper understanding later. Simply put, the style in which most of us were taught mathematics is not only inferior to others but also, quite possibly, detrimental.
The members of the Blake faculty are strong professionals and strive to build conceptual understanding as a matter of course. To this end, both Middle and Upper Schools teachers have voiced concerns about teaching for understanding through a traditional curriculum that emphasizes the inculcation of procedural skills at a breakneck pace. To increase our success at developing conceptual understanding in our students, we need both to research and share best practices in this area and to pare down our curriculum enough to create the space for richer investigations and discussions. Because of the way in which the math curriculum builds upon itself from year to year, identifying which topics need to be sacrificed for this greater good will require careful reflection by our math teachers across all grade levels. This is not a trivial process. Although conceptual understanding will have priority, we still need to manage the set of procedural skills students need for future success in school and at work. And we need to make these choices cognizant of the changing demands of the Information Age.
This quest for understanding is already underway. Upper School math teachers have already begun discussing how to focus the curriculum to better facilitate conceptual understanding. They have also started to investigate more widespread uses of Geometer’s Sketchpad to support greater exploration through problem solving in their classes. Middle School math teachers are re-evaluating the sixth grade curriculum to enrich it with deeper conceptual understanding. And Lower School teachers are refining their work to teach math facts by emphasizing understanding over rote memorization, to ensure that students are well prepared for Middle School and beyond. Highcroft’s third grade teacher Chris Passi has been an invaluable resource to Blake in this regard and is helping to organize a regional workshop on Cognitively Guided Instruction for one of our faculty summer academies.
All in all, this is a very exciting time to be teaching math, particularly at Blake. I invite teachers, parents and students to join in this reflective process through your questions and thoughts. Like Matteo Ricci, we are all teachers exploring the world. Unlike him, we are foremost aware of how much we have yet to learn.
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