Information Flow Control Method
Takayuki Yamauchi, Ph.D
Dept. of Math, VCSU
In this article, I will explain my content delivery method. Each class time of 50
minutes is divided into 5 to 8 short units, where each unit lasts from 2 to 5 minutes. 3 to
5 successive units constitute 1 block of lesson. The name Information Flow Control
means that the flow of information is limited to 1 unit (2 to 5 minutes long) at a time.
A typical 1 Unit consists of:
(1) A description of a definition or rule or theorem or formula with motivations behind it.
(2) A simplest possible example that reflects the essence (core idea or principle) of (1).
(3) An exemplary exercise that is slightly more complicated than the one used in (2) is
given to students.
(4) An exemplary exercise that is slightly more complicated than the one used in (3) is
given to students.
In (3), before moving on to the next example, while students work on the problem,
I walk around, and check their progress. When I spot an inactive student, I tell the
student to work on the problem. When I spot a stalled student, I point out what error has
been made, and tell the student how to correct the error. This procedure is repeated in
(4).
How I Construct Easy Examples
(1) A Typical Example Used in College Algebra or Applied Calculus Texts
I have not seen any math text that starts from a simplest possible example. For
example, in a section that covers how to graph a rational function containing 2 or more
vertical asymptotes, a typical text of Applied Calculus starts from a function having the
same structure as the following one,
1
x5
( x  1)( x  2)( x  4)
without showing the mechanism of a blow-up or blow-down near a vertical asymptote.
f ( x) 
Nobody can understand how to graph this function when no mechanism of a
blow-up or blow-down near a vertical asymptote is shown. The authors of the text are
expecting the readers to understand the subject without showing any method. This is
precisely why so many students told me, "Even though I spent 3 hours reading the text
before a class, I could not understand any of it.". Most math texts are written like this.
To avoid such waste of time, I tell my students not to read the text at all, but to keep it for
a future reference as a resource for those who have already understood the content.
(2) A Typical Example Used in My Class
I start from the simplest possible example.
Example 1:
(i) I write f ( x ) 
1
.
x
(ii) I review the definition of a vertical asymptote: the value of x that makes the
denominator 0.
(iii) I show how to graph this function near the vertical asymptote x = 0 as follows.
I let students evaluate this function at x  10 6 for x > 0 near 0, and at x  10 6 for x <0
near 0.
f (10 6 ) 
1
100

 100( 6)  106  1,000,000. (Blow-Up)
6
6
10
10
f (10 6 ) 
1
100


 100( 6)  106  1,000,000. (Blow-Down)
6
6
 10
10
Example 2:
(i) I write f ( x) 
1
.
x2
(ii) I tell the students to find the vertical asymptote, and evaluate this function at 2 values
very close to the vertical asymptote x = 2, x  2  106 , x  2  106 .
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Example 3:
(i) I write f ( x) 
1
.
( x  1)( x  2)
(ii) I tell the students to find the 2 vertical asymptotes, and evaluate this function at 2
values very close to the vertical asymptote x = 2, x  2  106 , x  2  106 , and at 2 values
very close to the vertical asymptote x = -1, x  1  106 , x  1  106 .
(iii) I expect that students will get stuck in (ii). They always get stuck. I let them
struggle for about 2 minutes. For example,
f (1  106 ) 
1
1
. They get stuck here.
 6
6
(1  10  1)( 1  10  2) 10 (3  106 )
6
(iv) I tell them that what matters here is not the exact value of the function at
x  1  10 6 , but the magnitude of the most dominant term in the expression. Adding
10 6 to  3 does not change  3 significantly. Introducing the approximate equality
symbol  , I show
f (1  10 6 ) 
1
1
1
 6
  106 (Blow-Down).
6
10 (3  10 ) 10 (3)
3
6
(v) I let the students handle the remaining 3 cases f (1  106 ), f (2  106 ), f (2  106 ) .
Example 4:
(i) I write f ( x) 
1
.
( x  1)( x  2)( x  4)
(ii) I tell the students to repeat the procedure of Example 3.
Example 5:
(i) I write
f ( x) 
x5
.
( x  1)( x  2)( x  4)
(ii) I tell the students to repeat the procedure of Example 4.
The Inverted Method and Mark Taylor's Method rely on class preparation by
students. The main flaw of these methods is that they have no back-up plan. That is,
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when a student cannot prepare for a class (due to an illness or an injury or a field trip or
an athletic event or a family obligation), the unprepared student cannot participate in the
class activity. Moreover, good on-line lectures are not always available. MIT, Harvard,
Stanford, etc., upload their lecture videos on YouTube. For example, MIT has the
highest admission standard in US. So, MIT lecture videos are all intended only for
exceptionally outstanding MIT students.
In my courses, absolutely no preparation by students is needed. If a student
wants to prepare for a class, the student can do so. But in reality, such preparation is
normally very time consuming, and is not feasible to busy students under many time
constraints. My methods are based on innovations. The methods used in Examples
1-5 shown above are applications of methods of Numerical Analysis. I also use methods
of Differential Geometry, Differential Topology, Geometric Analysis, Real Analysis,
Complex Analysis, Differential Equations, Mathematical Physics (they encompass my
research fields) to introduce new methods in my courses. I have taught Numerical
Analysis in 3 times before coming to VCSU in 2007. I have not seen any algebra text
that uses any tool from Numerical Analysis or Differential Geometry. I also found
that all the authors of the College Algebra texts and Applied Calculus texts I have
checked had no background in Numerical Analysis or Differential Geometry.
Because
of their absolute lack of knowledge in Numerical Analysis, they could not produce
any efficient method in their books.
One of the research problems I am working on now has its root in Numerical
Analysis. It is an unsolved problem published in a paper that appeared in The
Mathematics of Computation (a math research journal on numerical analysis) in 1990.
The problem encompasses 5 major fields of mathematics and Plasma Physics.
Conclusion
In order for a teacher to produce an innovative ground breaking teaching method,
the teacher must be always increasing the amount of background knowledge of the
subjects the teacher is in charge of.
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