“Growth is achieved by striking the right balance between support and challenge.”
The sentence peripherally addresses what my colleague has been arguing, that the balance between support and challenge has shifted over the last decade to require more support and less challenge. But, is this a bad shift? The article argues that current students, for a variety of reasons, are less resilient to failure and more needing of scaffolding and second chances to repair poor outcomes. And isn’t that in part a description of some of the students that arrive at campus with a different standing in terms of equity? Is this yet another tugofwar between rugged individualism and community support? Or is it really a warning that the butterfly of intellectual achievement is being damaged by external intervention in the struggle to emerge from the chrysalis?
And what about the changing role of the teacher? No longer a peripatetic scholar (if only from classroom to classroom) but a shepherd of bleating sheep?
And, can we ever find “the right balance between support and challenge?” This quarter I have been experimenting with test corrections and challenging extra credit assignments and I have seen the methods help some students to progress. But it is more work for me. (Another new reality of the information age: any challenging assignment in one quarter will not be challenging in the next quarter since it will be shared online. Challenges have a very short shelflife unless you go for those that are insolvable.)
I’ve said enough without really saying much of any real value. I’ll leave this week with a quote that should remind us as teachers not to take ourselves too seriously.
“The power of instruction is seldom of much efficacy, except in those happy dispositions where it is almost superfluous.”
The History of the Decline and Fall of the Roman Empire (Vol. 1, 1776; Vols. IIIII, 1781; Vols. IVVI, 1788) by Edward Gibbon, Volume 1, Chapter 4.
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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 standout; 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 semiclosed 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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A(0,n) = n + 1  If m=0 and n is a nonnegative 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, n1))  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 runtime errors occurred).
Two students realized that the answer would be a large integer, larger than a normal 64bit integer representation would allow (which explains the various runtime 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?
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I can best describe the course as a flyover 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 leastlike interaction with the emotional issues of learning; I mostlike 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.
]]>We were finding derivatives using the power rule. An exercise in the book asked for the derivative of
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.
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) allexponential form and 2) allfractional form.
Once the function is written as a summation of powers of x the determination of the derivative is easy.
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?”.
]]>“What is one of your most prominent memories of reading and writing when you were in college? Why do you think this memory stands out to you?”
I’m a math teacher, but I’ve always been a reader of novels (old and new). I read all the assigned texts in high school, thoroughly, and many others as followon—not extra credit, just interest. When I enrolled as a freshman at a Chicago community college on the southside of the city, I did not have a lifegoal in mind. My friends all went to fouryear colleges or universities, so I was adrift in a new sea. I was advised to take fundamental courses that would transfer. That’s what I did. I don’t carry many memories of the classes; they were in English literature, writing, analytic geometry, calculus, chemistry, philosophy (logic) and art history. I remember only dreamlike images of the instructors—the bus rides to and from class are more vivid in my mind. On one particular ride I saw a public service ad with a quote by Mark Twain. The quote was something akin to, “If you want to be a writer, write.” I don’t remember wanting to be a writer, but at that point in my life writing became another dish on my buffet of possibilities. I was eager to taste many experiences and experience many tastes (I couldn’t resist; we are covering commutative laws in my current math classes). With that preamble, I’ll answer the question.
My most prominent memories are of the writing course. I don’t remember the title of the course but we wrote short, fictional stories. I found that I liked to write and I liked what I wrote. I remember writing and rewriting on a small desk wedged into a large closet in the bedroom I shared with my older brother. I typed the handin version; no home computers then. I’m sure I never planned the structure of the stories; they just flowed from current topics. It was the time of the Vietnam War and I remember one of my stories set in a car traveling down a highway with a driver and one passenger, a hitchhiker. I no longer have the story so I only remember its tone and its ending. If I had analyzed it then I would have said that the hitchhiker was suffering from ‘combat fatigue,’ what today they call PTSD. I grew up in a bluecollar neighborhood, when that classification could be taken literally, and many of my neighborhood friends went off to the war—and returned damaged in many different ways. The story ended badly (both content and form I’m afraid). I never became a writer. Math took its place. But the exploration of trying to express emotion in words has stayed with me. My fears and uncertainties of life unfolding—it’s good to remember how empty and lost young students can feel.
]]>I am, in fact, the captain of two voyages right now: one to the land of continuity and change and one to the land of discrete computation. The inhabitants of the two lands are different species in the same genus and are related in the limit. We row past many cities on the shores of the lands and we take time between rowing sessions to enjoy the sites. Unfortunately we have too little time to become true citizens; we can only absorb some of the culture. It’s dangerous in fact to go too deeply into the interior; the lure of undiscovered treasure and the urge to map the wilderness can cause a crew member to mutiny and not return on the charted course.
But now I must assume my role as navigator and chart course corrections to give to myself as captain. The voyage must progress. It has to converge before infinity. After all, I have the ship owners to answer to on our return.
]]>Every time I prepare for a class I learn something new about what I already knew (don’t you love English and its homonyms). For example, I have taught on exponential equations many times, but in reviewing for a review of the topic I marveled at how exponentials, well before the long run, magnificently outpace polynomials. Nothing really profound—I was comparing a simple quadratic function and a simple exponential function, the two topics for the lecture.
I used a TI84 for demonstrations so I’ll include some screen shots.
Here we have the two equations.


The first graph and its window settings is shown below. The leftmost curve is the quadratic. The quadratic seems to handily outpacing the exponential. The graph was created with the ZOOM 1^{ST} QUADRANT command.  Increasing the height of the viewing window, by about a factor of four, shows that the exponential is catching up to the quadratic (below).  Increasing both Xmax and Ymax, but only doubling Xmax, it becomes very clear that the exponential leaves the quadratic forever behind starting at about 10 (below). 
Admittedly, some of the effect of the graphs relies on the shallowness of the first two windows, but it still is exciting to see the power of an exponential growth. Whenever I teach about exponentials, I try to emphasize to my students that any use of an exponential growth model must require the determination of a practical domain—going too far is going too far.
Since I’ve shown some calculator graphs, let me reflect on the use of a graphing calculator, such as the TI84, in a math class. Why should an instructor require a graphing calculator when there are so many more visual, easier to use, graphing packages available for free on the internet? Here are two reasons.
First, the TI84 is a readilyavailable, commondenominator tool. Many students already own one, or can borrow one, and for those students who don’t own one, there are usually a pool of them available at the school library. The calculator has sufficient capability to draw graphs of equations or create equations of scatter plots. And it is not hard to master the basic commands. As an alternative, requiring all students to bring a laptop, tablet or phone with appropriate app is not always possible. And not every community college class can be equipped with studentavailable computers.
Second, a calculator is generally needed on math exams to help with at least arithmetic operations on decimals, fractions, or irrational numbers. If the course is also teaching regression of data points to best fit curves, this level of calculator is also needed for exams. Allowing students to use a laptop utility, or their phone, with substitute software, is just too much temptation for students to move beyond graphing/calculation software to symbolic manipulation software. This is also a reason to not use higherfunction, symbolic manipulation calculators instead of graphing calculators.
That’s my reflection for the week of January 9, 2017.
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Q1. Create a piecewise function that represents the graph shown below. Assume the pieces of the graph are built from the functions listed below, with appropriate transformations:
y = x
y = x^2
y = ln(x)
The graph shows a continuous, piecewise function from 0 to 3 that was build from three joined segments of subfunctions,
y = x from 0<=x<=1
y = (x2)^2 from 1<x<2
y = ln (x1) from 2<=x<=3
Those three lines are, essentially, the answer I was looking for.
Most of the students answered the question with transformations, not always correct transformations, but in the spirit of my intention. About 1/3 of the students, however, answered as follows (some with mistakes in the inequalities, but let’s ignore that),
y = x from 0=<x<=1
y = x^2 from 1<x<2
y= ln (x) from 2<=x<=3
From the reading of the instructions, they seemed to have assumed that ‘with appropriate transformations’ meant that the transformations did not need to be shown in the list of piecewise functions. Apparently they thought that I was testing for the correct partitioning of the domain, and that I didn’t really care about an accurate representation of the graph for each partition. Since more than just one or two students chose this interpretation, I am wondering if perhaps the question is indeed ambiguous. Therefore, I have a question to those of you who feel that you can put aside what I’ve already explained and can read the question as if you were taking the quiz along with my students. Do you believe that the question, as worded, is ambiguous and might have misled you into replying without transformations? Do you have any suggestions for a better rewording of the question? Comments at all?
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P(h)={
10 if 0<h≤1
10+8 if 1<h≤2
10+8+5(ceiling(h)2) if 2<h≤24
}
To find the charge for 4.5 hours the function would be evaluated as follows,
0<4.5≤1 is false
1<4.5≤2 is false
2<4.5≤24 is true
therefore,
f(4.5) = 10+8+5(52) = $33
The piecewise function has two parts, the rightmost part that specifies qualifying conditions and the leftmost part that specifies a dedicated function to evaluate, if its associated condition is met. That is, the unique condition in the righthand part, that evaluates to true for the input to the function, identifies its corresponding dedicated function in the lefthand part as the appropriate function to evaluate the input. In this case, as shown above, the condition that is true given 4.5 as an input is the condition in the last line of the function specification. Therefore, the function is evaluated with input 4.5 using (10+8+5(ceiling(h)2)) as the relevant dedicated function, returning an answer of $33.
The details really are not important, except as a setting for my reflection.
The first worksheet presented several definitions of piecewise functions, similar to what was just shown. For each of the functions the students needed to graph the piecewise function and then evaluate the function for several input values, both algebraically and graphically. We spent approximately 15 minutes before the exercise going over how to graph piecewise functions on a TI84 calculator so the students were testing that new knowledge at the same time as they were testing their understanding of the evaluation of piecewise functions. All of the students participated, some in small groups, some alone. I wandered the classroom looking over their shoulders and answering questions. They appeared to all be in search of answers to the worksheet problems. From the questions I received from some of the students, it was clear that not all of the students had really understood my explanations of piecewise functions, even with examples still on the board. This has become a common enough result that I am convinced that in my classroom of 40 students, from so many different backgrounds, there will always be some students that will not understand my lecture in the way that I think they should. More individual attention will apparently always be needed for some students. The inclass worksheet approach seems to provide a good way to identify at least some of the students in need.
The second worksheet presented two application problems, one about agebased pricing at a buffet restaurant and the other about the time (in minutes) that a person may safely scuba dive at a certain depth without having to decompress while surfacing—depth in feet. The second worksheet was an inclass project on the day after the first worksheet project. I lectured on some examples of application problems before distributing the second worksheet. The students worked with interest but again I received questions that showed a lack of understanding of what I had explained in the lecture. To be sure, most students had no problem establishing the correct form of the piecewise functions for both problems, but almost all of them had some trouble in accurately completing each form so that each result was a piecewise function and not just a piecewise relation. What that exactly means is not important here.
In summary, I think that each worksheet was a good, active learning experience for both the students and for me. And again, the inclass worksheets made me confront the reality that no matter what information I think I am transmitting in my lectures, not all students are receiving the intended message.
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