Van De Walle, Ch. 3 The importance of understanding

Reflection:

Why does it matter whether or not someone understands mathematics on a conceptual or relational level (lots of linkages) versus a procedural or instrumental level (an emphasis on rote memorization and drill)?

The broader a person’s mental schema, the greater the number of neural connections, the quicker the processing speed.   This is fact!  Neuroscientists have observed that neurons that fire together wire together. (The Brain Which Changes Itself, Norman Doidge. MD, pp. 63-70) 

Too often, teachers do not systematically link procedural knowledge to underlying concepts.  When the brain processes new information in isolation, new information doesn’t readily transfer to long-term memory.

Neurons that fire apart wire apart – or neurons out of synch fail to link.  (Doidge, p.64)

Dr. Van De Walle’s analysis is consistent with my own personal observations of “slow learners”.  For the field of education, the ramifications of Dr. Van De Walle’s argument and what modern neuroscience is telling us are mind-blowing:

A negative effect of the behaviorist influence on education was the fragmenting of mathematics into seemingly endless lists of isolated skills, concepts, rules, and symbols.  Each has to be mastered before moving on.  The lists grow so large that teachers and students become overwhelmed.  When items are learned rationally, they become part of a larger web of information.  Frequently, the network is so well constructed that whole chunks of information are stored and retrieved as single entities rather than isolated bits.  (Van De Walle, 27)

Recently, I spent a day as an Instructional Assistant Substitute teacher, and worked with a 3^rd grade math student with a learning disability.  She had learned procedures for multiplying 3-digit numbers by a 1-digit number and  used a multiplication table for math facts she didn’t know to solve problems.  When given story problems, however, she frequently added the factors instead of multiplying them, she didn’t consider whether her result should be a big number or a small number, and lacked strategies for visualizing the problem.  In one problem, I modeled a problem as a repeated addition problem.  Rather than multiplying the digits, four 4’s for example, she counted on her fingers – a highly inefficient strategy – which led to errors in how she applied the borrow-and-trade procedure.  I modeled another problem involving how far a plane traveling 367 miles per hour would fly in four hours by drawing a diagram, which made it evident that we had another repeated addition problem.  The little girl was capable of using strategies for understanding, but it seemed to me that she may not have been exposed to them.  By the 4^th problem, she bypassed the repeated addition step on her own and applied the multiplication procedure appropriately.

In education, perhaps more-so than any other field, people are demanding results.  What can be more important than the future of our country?  Dr. Van De Walle cites research that seems to indicate that despite a back-to-basics, renewed emphasis on traditional computation skills since the 1970’s, student problem solving skills and concept knowledge are not significantly improving.  (Van De Walle, 27)  Despite huge educational budgets, and an ever-increasing emphasis on “data-driven” instruction, despite general acknowledgement that differentiation is an educational right, huge existential questions remain unresolved.

What are our mathematics assessments truly measuring?  Is our educational pacing developmentally appropriate to individual learners?  Is our mathematics instruction preparing learners for the thinking requirements of the 21^st century? How badly does a bias towards "answer finding" skew our educational data?

For years, we were told the stock market could only go up.  Then, we discovered that our gains were built upon a phony edifice.  An entire banking system came to rely on derivatives from an inflated real estate market, where properties were "flipped several times over".  Then our banking system collapsed, only to be rescued by a multi-billion dollar bailout.  Education has come to rely on thinking from the results-driven business world, and I wonder whether the problems identified by Dr. Van De Walle have gotten worse because of a misplaced emphasis on this thinking, not better.