Calculation

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Calculation

• March 23, 2004

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For today

• Student reflections
• Some remarks on calculation

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Adult motivation True? Why?

• Most of my work has been with adults in
  developmental math classes. Many do not know
  their multiplication tables, and rely heavily on
  calculators for assistance. Because they don't
  know the multiplication tables, every problem
  takes much longer to solve. Dr. Ginsburg
  suggested that if the children were not taught
  the multiplication tables, they would want to
  learn them because it would make solving problems
  more efficient. I have not seen this motivation
  in my adult students.

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More adult experience

• When the equations were flashed on the screen, I
  immediately knew that it was hopeless to remember
  them and they had no relevance to my life so I
  began thinking about what I would have for
  dinner.

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When I was younger

• I remember sitting at the table memorizing
  everything from my times table to spelling words.
  I personally did not enjoy this task of
  memorizing, but I did enjoy being able to finish
  a test faster than the other children and winning
  many spelling bees, so what's so wrong with
  teaching facts before the procedure? Look at the
  people in this class, I bet most of then learned
  this way and look how far we have all come.

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Another memory

• In grade school, I was stuck on 7 x 3 and since
  we weren't given scrap paper to solve the
  problems, I drew seven columns and 3 rows of
  little dots. The teacher told me I was not
  allowed to doodle and when she realized what I
  was doing, she said "and definitely no cheating"
  in front of the whole class!

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Parent involvement? Part 1

• Parents of students who are struggling to
  memorize their "doubles" and "doubles plus ones"
  facts come to me seeking extra work sheets for
  their children to complete at home as extra
  homework. I try to explain to these parents the
  importance of understanding how to get a
  particular answer, not simply being able to spit
  out the correct number word, yet they just dont
  seem to understand.

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Part 2

• I observed a disturbing incident on the subway
  this week. A mother was drilling her child about
  number facts, apparently from a homework
  assignment. The little girl seemed so flustered
  and when she got the answer wrong (she said
  16-813), her mother belittled her loud enough
  for most of the people in the subway car to hear.
  Everyone around just shook their heads.

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Clinical interview doubts

• I question the validity of clinical interviews
  because people do not always accurately report
  what they did. For example, there is no way of
  "knowing" whether the child did really have a
  picture of the carrots in his head (since we
  cannot go inside of the child's head). I also
  question whether "thinking" can be studied
  because we can only study one's thought process
  through the information that they report to us,
  and we have no way of determining whether the
  self-report is reliable and valid. Thus, how can
  we utilize the clinical interview to help us
  scientifically devise better methods for teaching
  children math?

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Applied Behavior Analysis

• I was disheartened by the discussion of the
  "Applied Behavior Analysis" group during
  yesterday's lecture. It seemed to be a label for
  a group that separated them from the cognitive
  camp, or any other theory like an "us" vs.
  "them" discussion. But why choose?

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Calculation

• Overview of coping with school math
• assimilation (existing knowledge)
• accommodation (do what they say)
• Radical point there is existing knowledge

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Five ways to calculate

• Do routine correctly without thinking
• Do routine correctly with full understanding
• Invented strategies
• Bugs
• Sloppy stuff slips

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The first two

• Follow the routine without understanding
• Student comment Learning calculations can be
  like learning number facts
• This is fairly standard priority of
  accommodation
• Undermines gold standard of testing, the correct
  answer
• Full understanding next week

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Invented strategies

• Lots of examples. Bob Davis 22 18 (-6)
  (10) 4. You saw some last week in connection
  with number facts.
• Students point out invented strategies are
  syntheses, not complete innovations.
• Part assimilation, part accommodation. A good
  balance. Implies understanding?
• The algorithm is a privileged invented strategy

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Vygotsky says
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How to assess invented

• Blackboard Math example
• What is the math in this?
• Individual clinical interview

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What to do with invented?

• A matter of value and educational policy
• Stamp it out
• Ignore it
• Build on it-- student comment
• How would you do that?

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Bugs

• Example of Diane (video)
• A slightly or seriously deranged invented
  procedure
• Priority of assimilation? Origins
• Associated with belief Tammy (video)
• Cynical and/or reflecting what was taught?

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What to do with bugs

• A matter of value and educational policy
• Stamp it out
• Ignore it
• Build on it-- student comment A lot can be
  learned from mistakes
• How would you do that?
• Buswell and John explaining error of their ways.
  
• Project ask students to explain others bugs

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Slips

• Neatness is a virtue
• Neatness is boring
• Too much can stifle creativity
• Too little can mess things up

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Reminder

• Project proposals