Blocked

I’m having trouble getting started on my blog this week.  Coincidentally, I just read the blog, “Starting Trouble,” by &.  It explores the similarities between writer’s block and programmer’s block.  I wonder if the article inspired me.  One tonic to relieve the block, suggested in the blog, was to just write, anything at all if necessary, even if gibberish.   So here it goes.

I am tired tonight.  I spent several hours today preparing worksheets and reviewing a lesson plan on the topic of optimization in calculus.  Optimizing is finding a value that minimizes or maximizes a model (mathematical function).  A great deal of vocabulary surrounds the procedure for finding the values.  I found it difficult to get to the heart of the matter after spending much of my time defining terms.

The minimum and maximum values on a particular domain are values of a function that stand-out; they are extreme values that are notable by being higher or lower than any other values in their neighborhood.  What exactly is a neighborhood?   Informally, a neighborhood is an interval around an independent value, usually a small, symmetric interval, as small as you need it to be to prove a point (which in this case is to identify a point).   An extremum can be a relative minimum or a relative maximum, also called local minimum or local maximum, the connotation of local fitting with the terminology neighborhood more closely than the word relative, but relative gives the sense of judgement tempered by location, so it too suggests neighborhood.  When you have more than one extremum, you have extrema; English showing its roots in older languages.  Anywhere you are to find an extremum you must also find a critical point or an endpoint (but finding a critical point or an endpoint does not guarantee an extremum).  Critical points come in two flavors, singular or stationary.  Endpoints are ending points of closed or semi-closed intervals.  Singular points are internal points (not endpoints) where the derivative of the function evaluated at the point is not defined (then how did you evaluate it you might ask).  Stationary points are internal points where the derivative of the function evaluated at the point is zero (hence the idea of stationary since the derivative is all about rate of change).  The absolute maxima or minima, which in the general case may not exist, are relative maxima or minima which have the highest or lowest values of the function, respectively.  But they are guaranteed to exist if you constrain the function to continuity and the domain to being closed on both sides.  The name of the guaranteeing theorem is the Extreme Value Theorem (back to that term again).  You test critical values to see if they are relative minima or maxima by looking at sign changes in the derivative function from one side of the critical point to the other (they are internal so they must have two sides).  The endpoints, where they exist, must be tested individually in half neighborhoods, usually at the end of the procedure, but that is not why they are end points, as was noted earlier.

That should be enough to get started, certainly enough to meet the mark of gibberish.  Unfortunately, this gibberish is what must be gibbered to introduce optimization using techniques of differentiation.  Is it any wonder that students develop a mental calculus in the study of calculus?

 

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Extra Credit

In my discrete mathematics class this week I challenged my students with an extra credit task.  Write a program to compute A (4,4) and print the decimal digits of the answer.  The original version of the function was developed by the German logician Wilhelm Ackermann in the 1920s.  The Ackermann function is a deceptively simple recursive function with a not-so-obvious, very-high rate of growth.  It is defined as follows:

  A(0,n) = n + 1 If m=0 and n is a non-negative integer
  A(m,0) = A(m – 1, 1) If m is a positive integer and n=0
  A(m,n) = A(m – 1, A(m, n-1)) If m, n are positive integers

 

As an example, a call of A(1,1) would result in the following calculations.

  A(1,1) Start
  A(0, A(1, 0)) m, n are positive integers so the third line is evaluated.
  A(0, A(0,1)) A(1,0) evaluates to A(0,1) on the second line.
  A(0, 2) A(0,1) evaluates to n + 1 = 2 on the first line.
  3 A(0,2) evaluates to n + 1 = 3 on the first line.

Not bad.  All students in discrete mathematics are required as a prerequisite to have passed a Java or C++ programming course, so assigning a simple program is a reasonable request.  In fact, programming this function is a ‘walk in the park.’  The program is essentially an evaluation to an integer when m  = 0 and two recursive function calls when m is not 0.  All that needs to be added is scenery:  the syntax for a function declaration, data definitions, conditionals that implement the decisions on the function calls, and a print statement to display the answer.

I was vague about how much extra credit the assignment was worth, but I did tell them that points earned would be added to their graded homework tally—a nice way to win back some lost homework points.

Five students rose to the challenge; some standing taller than others.

Two students wrote a simple program; tried to run it for A(4,4); and gave up when nothing printed (various run-time errors occurred).

Two students realized that the answer would be a large integer, larger than a normal 64-bit integer representation would allow (which explains the various run-time errors).  The students decided to use a Java BigInteger class that allows for very large integers.  Doing so allowed one student to print the answer for A(4,2), a number with 19,729 digits.  But the A(4,4) failed with cryptic errors.  The other student divided the computation into parts that could run on multiple processors but had no answer displayed, even after the program ran all night.

One student did not write a program but analyzed the mathematics behind the recursive calls to compute A(4,4) in the form of an approximate exponential , 2^(2^(2^65536)).

A small challenge; but one that was never going to be met.  No one was going to print all the digits of the answer to A(4,4).  Abuse of trust?  I hope not.  I gave credit for the programs and more credit for the realization of the need to address the size of the answer; and even points for analyzing the mathematics instead of writing a program.  Students like to be challenged (at least some students) and since there was no big prize attached to the challenge, and I gave consolation prizes, I do not think the students felt cheated (none said so anyway).  I do not often use extra credit as a tool in teaching.  This week told me it might be worthwhile to use it more often, at least once in a while.  What do you think?

 

Learning to Teach

I attended a very interesting four-hour training session today on the basics of teaching.  (A fact in the teaching profession: college teachers are qualified on content credentials, not teaching; much of ‘teaching’ is learned in the classroom, in office hours, in special training, or in conferences with colleagues and researchers.)

I can best describe the course as a fly-over of a very large territory with a knowledgeable tour guide who has spent many years exploring and mapping the terrain.  Some of the features of the topography:  varieties of intelligence; preferred styles of learning; the symbiosis of teaching and learning; cultural influence and interaction; metacognition; and emotion in teacher and student, to name a few.  The course practiced what it was teaching and taught in segments of “look below, over there,” “listen to this tale and the voices of fellow travelers,” and “walk about to stretch your minds and seek other opinions.”   (Yes, appeals to visual, auditory and kinesthetic learning styles.)

What did this teacher/student enjoy the most?  The filling out of worksheets to help identify preferred learning styles.  The task was directed, focused, nurturing, engaging, connected and valued.  Why? Because it was about me!  What could be more interesting to learn?  Everyone knows that somehow the best teachers facilitate learning.  That learning is a construction in the nervous system that allows new pattern recognition.  (No, I can’t point to research; this is a blog; I can make unsubstantiated generalizations.)  That pattern recognition allows the construction of new pathways to more pattern recognition.  A recursive build of patterns.  (Serendipitously, something we covered, briefly, in my discrete mathematics course this week.)

The purpose of the training is to teach the teacher that there is more to teaching than teaching and so bring us back to land with the firm intention of trying something to change our teaching to be more sensitive to the learning of the students we teach.  Some change should be chosen in the area of our learning preferences that we least prefer.  For without coaching, we teach the way we like to learn, and our students must wear the shoes that fit us well.  I least-like interaction with the emotional issues of learning; I most-like analysis of systems of symbols and logic.  I need to feed the emotional beast?  Hopefully, it will not eat me for dinner before I learn to tame it.  Will I succeed?  I have to go to analyze the situation first.

When Will I Ever Use It?

All math teachers have heard the questions: “Why do I need to learn algebra?”, “When will I ever use it?”.  In my calculus class this week both questions were answered.

We were finding derivatives using the power rule.  An exercise in the book asked for the derivative of

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At this point in the course the function needs to turned into a summation of powers of x and the power rule needs to be applied to each resulting  term.  When the authors created this problem, intentionally or not, they mixed two styles of algebraic notation.

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The function definition mixes together both an exponential form and a fractional form.  The best approach in solving this problem is to change to a common form and then distribute multiplication over addition. There are two options, 1) all-exponential form and 2) all-fractional form.

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Once the function is written as a summation of powers of x the determination of the derivative is easy.

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The students worked on the problem for a few minutes and most of them used Option 2,  fractional form, to solve it.  I used Option 1, exponential form.  I wrote my solution on the board at the end of the exercise.  Surprisingly, there were two students who could not understand why my solution worked.  One student was rusty in his algebra skills (gleaned from earlier work with him) so his confusion was understandable.  The other student had participated in class discussion and answered open questions on other sequences of algebra, so his confusion was more unexpected.  Not understanding and misunderstanding in math can be difficult to correct.  The student is seeing something different from the instructor, misinterpreting symbols, or missing unstated assumptions.  The student, by definition, cannot tell the instructor what is not seen or is missing.  The instructor, being fluent in the operations, does not see what the student is seeing.  An inefficient dialogue can be the result.  In this case the misunderstanding was resolved for both students by reminding them of the basic rules of exponents in algebra.  A simple problem, but a problem that gives answer to the leading questions: “Why do I need to learn algebra?”, “When will I ever use it?”.