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"We are called to be architects of the future, not its victims."

"Make the world work for 100% of humanity in the shortest possible time through spontaneous cooperation without ecological offense or the disadvantage of anyone."

- Buckminster Fuller

Wednesday, February 17, 2010

Van De Walle, Ch. 2, What it means to know and do math

Notes and Reflections on Elementary School Mathematics by John A. Van De Walle

Chapter 2:  Knowing and doing math

Reflection: contrast the new and old paradigm:

Dr. Van De Walle describes Math as a science of pattern and order (a reliance on reason, relevant tasks, students defend their reasoning)

          “The world is full of order and pattern” (Van De Walle, 7) – all of the great ancient cultures discovered that we are immersed in a world of pattern and order.
          My favorite example of mathematics revealed as a science of pattern and order comes from the movie, Donald Duck in Mathmagic Land.  I remember how I felt about math when I first saw it in elementary school in the 1970’s.  Students continue to ooh and ah at patterns revealed in the movie whenever I show it.
Mathematics discovers this order and uses it in a multitude of fascinating ways, improving our lives and expanding our knowledge.  School must help children with this process of discovery.  (Van De Walle, 8)
          Consider the story of Friedrich Gauss, who was tasked by his teacher to compute the sum of a sequence of numbers.  After solving the problem in mere seconds, Gauss was accused of cheating rather than complimented for his brilliance (http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss).
          Math Investigations explores the idea of volume in a developmentally appropriate way; unfortunately, it doesn’t match the pacing guide.

The traditional view of mathematics as a collection of rules and procedures (a reliance on authority, students are taught arbitrary rules, and students learn to rely on memorization)

          No school wants to be held before the public as a failing school; students who can’t pass standardized tests reflect poorly on the teacher and the school; thus, lesson plans and testing data are scrutinized to verify that students are keeping pace.
          Computers compute better than people do; by 2030, according to renowned futurist, Ray Kurzweill (http://www.kurzweilai.net/index.html?flash=1), $1,000 of computing equipment will have the same computing power as the human brain – old skills are indeed becoming obsolete.
          America was built on a “frontier” mentality which valued thrift, hard work, sobriety, and a can-do spirit.  America was known for its can-do practicality.  Where is that same frontier mentality today?  Due to pacing guide pressure, when jobs are on the line, educators flee activities that involve deep level processing for the kind of one-size-fits-all procedure/rule driven instruction opposed by Van De Walle.
          Do we limit opportunities for higher level thinking to the gifted?  Or do we allow children of poverty the same opportunities to discover order and patterns in mathematics?

Reflection:  becoming mathematical problem solvers is, in my opinion, the most important of the five Curriculum Standards goals for students highlighted in this chapter

1.       learn to value mathematics
2.      become confident in their ability to do mathematics
3.      become mathematical problem solvers
4.      learn to communicate mathematics
5.      learn to reason mathematically

          An environment conducive to problem solving involves worthwhile tasks, builds confidence, requires students to elaborate, and builds reasoning skills – thus, goal # 3 incorporates the four other goals.
          In this chapter, Dr. Van De Walle provides 5 examples of worthwhile tasks, and provides a play-by-play analysis of how these tasks might play out in a classroom environment.
          Activities such as Start and Jump, Combining Tiles, In-between numbers, Finding Areas, and One equation (Van De Walle, 11-17) encourage students to notice patterns, record observations, and reflect upon what happened when they varied their approaches – student processing time is built in to the process, which should facilitate the development of long-term memory.
          Dr. Van De Walle contrasts true problem-solving with mechanical answer finding; the difference is in the level of thinking being tasked; problems without a single clear answer involve analysis and reasoning; students become active learners instead of passive recipients of base-line computational knowledge
          Are we creating a generation of struggling learners who are being advanced through too quickly and in a manner that doesn’t allow enough time for core understandings to develop in their long-term memories?

Reflection:  overcoming a reliance on the “math god” – what is the single most important thing that can be done to prevent this concept from developing?

          Students are taught to be passive learners of mathematics from a young age; rules and procedures can seem arbitrary, with answers, rules, and procedures passed down to students from intelligent authority figures
So where do these rules come from?  Is there a “math god,” someone who has all the rules and figures the out and to whom the teacher is somehow connected?  In such an environment, what else could children come to believe? (Van De Walle, 9)
          Teaching for understanding, on the contrary, empowers students to discover order and pattern for themselves
          When students are habitually expected to elaborate their thinking as problem solvers, only then can learned helplessness be overcome

Reflection:  Why does Lauren Resnick think mathematics might be characterized as an ill-structured discipline rather than a well-structured one? (Van De Walle, 17)

          Most teachers are control freaks by nature; this element of Math Investigations especially infuriates many teachers and parents of struggling learners – “but how will little Johnny pass the test?”
          A lack of structure causes frustration, which shuts down learning, especially for a generation of students who have grown accustomed to having and endless succession of rules and procedures handed to them
By fifth or sixth grade there are many children who simply refuse to attempt a problem that has not been first explained:  “You haven’t shown us how to do these.”  This is a natural consequence of the bits-and-pieces, rules-without-reasons approach to mathematics.  Children come to accept that every problem must have  a method or solution already determined, that there is only one way to solve any problem, and that there is no expectation that they could solve a problem unless someone gave them a solution method ahead of time.  Mathematics viewed this way is certainly not a science of patterns and order.  In fact, it is not mathematics at all.  (Van De Walle, 9)
          The sentiment expressed by Resnick is unfortunately worded and unnecessarily inflammatory, in my opinion.  I agree that a reliance on mechanical answer finding is precisely what’s wrong with mathematics instruction, but that is not the same as saying there is too much structure structure.  Learning through explorations of open-ended questions, probably requires more up-front structure that traditional methods.  A well-designed guided math component, analogous to what Jan Richardson has done for guided reading, would enable the teacher to assume the strategic question-asking, discourse promoting, evaluative role Van De Walle envisioned.