Calculation

March 23, 2004

For today

Student reflections

Some remarks on calculation

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.

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.

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.

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!

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 don’t seem to understand.

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 16-8=13), her mother belittled her loud enough for most of the people in the subway car to hear. Everyone around just shook their heads.

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?

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?

Calculation

Overview of coping with school math: assimilation (existing knowledge) accommodation (do what they say)

Radical point: there is existing knowledge

Five ways to calculate

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

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

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

“...the development of the child's spontaneous conceptsproceeds upward, and the development of his scientificconcepts downward... The inception of a spontaneousconcept can usually be traced to a face-to-face meetingwith a concrete situation, while a scientific concept involvesfrom the first a 'mediated' attitude toward its object... Inworking its slow way upward, an everyday concept clears apath for the scientific concept and its downwarddevelopment ...[giving it] body and vitality. Scientificconcepts, in turn, supply structures for the upwarddevelopment of the child's spontaneous concepts towardsconsciousness and deliberate use.” Vygotsky Thought AndLanguage, Kozulin: P.193-94

Vygotsky says

How to assess invented

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

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?

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?

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 other’s bugs

Slips

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

Reminder

Project proposals