(Bead 6. .)                   Send note to Paul Prueitt .                      (Bead 8. .)


A Question of Access


Part 2: Chapter 7

Re-teaching arithmetic using number bases other than base 10

August 15, 2002



Purpose of the chapter


The Nodal Forest Learning Strategy asks the learner to list the topics that are in a specific curricular unit.  The curricular material is defined to be that unit of material that is the next learning task.


The learner sorts these topics into two categories. 


New topics may be added to the categories.


Topics may move from one category to the other category. 


In fact, some students will list a topic as “known” and then later come to understand that the topic is really “not known” and perhaps even come to understand


1)        That he/she has a specific deception that is imposing confusion experienced during the taking of a test.

2)        That the topic was not understood in a way that is characteristic of a specific confusion that he/she has about the topic, or even more generally the nature of abstraction, arithmetic or the non-arithmetic parts of higher mathematics.


Remember that the first topic category is to be rehearsed, but the second topic category is to be left alone unless the student feels an interest in one of these topics. 


As mentioned in Chapter 6, cognitive neuroscience and academic theories of self (autopoiesis) suggest that an inhibition of natural interest has been effectively turned on by the learner’s experiences and by the reinforcing effect of the ubiquitous social philosophy supporting mathematics dislike.   How can this natural interest inhibition be turned off?


Various exercises are provided to turn off this inhibition if the inhibition is not critically sever.  However, with as much as 60% of entering freshman, the inhibition of natural interest is systemic and unrelenting.


So can anything be done to fully remediate the learning capability and interest in mathematics? 


Is this remediation always possible even with adult learners?


The author’s claim is that yes remediation has been shown in a high percentage of the student that he has taught.   Moreover, the conjecture is that a web based learning system can be used to provide a low cost relief from the mathematics anxiety that most adults have learned to live with. 


Generalization of the learning system can be used to teach other subjects and in general knowledge management processes in business and education.


The process must also allow the experience of some knowledge about cognitive processes and the image of self. 


In addition to a process model of three states of student orientation


{ motivated, bored, fearful }


A second process model is discussed


{ easy to understand, difficult to understand }


Stories from the Author’s collection of student papers are to be included in this chapter and then the strategy used at Saint Paul’s College by the author in 1994 will be developed. 


This strategy involves the learning about the Nodal Forest Learning Strategy, and then the use of this strategy in the re-introduction to the students of the fundamental notions of counting numbers, including addition, multiplication and division. 


Some history of the counting number is given in Chapter 10 (Part 1). 


In cases where the entire class is faced with learning arithmetic as college freshman, the behavioral problems of students can be a factor. 


Again, the use of case history helps to set up the context in which the re-introduction of arithmetic is made.  The 1994 study showed that the class would periodically move between the motivated, bored and fearful states.  Individuals would also move between these states.


The instructor can be aware of the state transitions.  Bored and fearful class-states can be converted into a completely motivated state, as expressed by the entire class. 


Again some stories help to illustrate. 


One young man attended for the first three days of a college arithmetic class.  He was keen on motivating the class as a whole to mis-behave in every way possible.  Then he did not attend class for two weeks.  Meanwhile the class had been challenged to learn to add and multiple in any base { 2, 3,  4, . . . , 25 } and to check the answer by converting the beginning of the problem and the end of the problem to base 10.  Some interesting theorems are actually discovered by these students, and the novelty was so great that some students began to attempt to engage the upper level mathematics majors in “the issues”.  {This is another story.}


The young man came back into class just as the class was struggling with a new bombshell that the author dropped.  The class was asked to multiple two fractions when the fractions where in an arbitrarily selected base (other than 10).  The class moved immediately from motivated to fearful.  In fact the class was pretty upset. 


The young man was genuinely pleased with this state of the class.    But the class accepted the challenge, and this seemed to be sufficiently interesting to the young man that he returned the next class period. 


The beginning of the next period was remarkable.  The students were mostly ordered and attentive.  The young man was talking with several others.  But it was clear that he did not understand what was going on in general.  When the author walked in, he kept attempting to organize a student walk out to protest the unreasonable demands that had been placed in the class. 


For a minute or so there was some tension in the class, and between the young man and the author.  However, several of the other students turned on him and made it clear that his behavior was not acceptable.  This actually seemed to startle the young man.  He sat back and observed, and in his attentive state he was deeply changed. 


All most all of the students knew how to multiple factions in arbitrary bases, and the few who had not figured out the procedure were very quickly in line.  The class even accepted a much more, actually radically more, difficult challenge of adding two fractions in base 17.


After the students left the class, the young man stayed behind.  A discussion followed and some agreements where made; to forgive the absences, to allow the previous grades to be replaced with the average of all grades from then on out, etc.  This man made an A in the course.  Later that year, he and the author meet one day and sat down and talked about the experience. 


The full story will have to be told with some care.  The author’s racial background is English, Irish and Russian.  Saint Paul’s College is a small Historical Black College in rural South-central Virginia.  His contract to teach arithmetic classes for two semesters was controversial for a number of reasons.



These notes are formative towards Prueitt’s new book on learning theory.

We expect that the book will be developed over the next four months.

(Bead 6. .)                         (Other Beads)                         (Bead 8. .)