Deltas and Epsilons

Here is a way to help your students understand deltas and epsilons. As I've said before I'm not that sure that being able to find the d-e relationship alone does that. So lets assume students are able to compute d-e in general and look at the lim(x-->3) (2x-5) = 1 where the usual approach will give d = e/2

 

Begin by drawing the "box," that is the rectangle whose vertical sides are x = a - d and x = a + d and whose horizontal sides are y = L+ e and y = L - e. Use a calculator or better yet a good graphing program like Winplot (or Geogebra, Nspire, etc.) Use sliders for the actual values of d and e.

 

 

CASE 1: With d = e/2 the box will be twice as high as it is wide. The graph of y = 2x - 5 will go exactly through the top left and lower right corners of the box. For all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. The requirements of the definition are met and we have proven that the limit is 1.

 

CASE 2: Let d = e/4. The box is now 4 times as high as it is wide. The graph enters and leaves through the sides of the box. For all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. The requirements of the definition are met and we have proven that the limit is 1.

 

CASE 3: Let d = e, the box is now a square. The graph enters the bottom of the box and leaves through the top. It is NOT true that for all values of x between 2 - d and 2 + d the corresponding function value lie between 1 + e and 1- e. There are some points outside the box top and bottom that should be between the horizontal lines but are not. The requirements of the definition are NOT met and we have NOT proven that the limit is 1.

 

It makes no difference what value of e is used. And this is important too. The fact that  the box looks the same (technically a similar figure) for a particular delta is the graphical way of saying "for any e" ("for all e's", "for every e"). Large e, small e, the d = e/2 box is always twice as high has it is wide and the graph goes through the corners. This makes it easy to believe that as you go smaller and smaller the picture will not change ever.

 

With the sliders you can play with different d-e relationships for this or any function even if you cannot find algebraically the largest such relationship.

 

For polynomial and other functions where  the relationship is something like d = min(1, e/3) the CASE 1 graph is not possible. The function may enter or leave at one corner but not the opposite corner; it will leave through the opposite vertical side meeting the requirements; it's more like Case 2. But with small values of d an e local linearity takes over and things look like Case 1.  Taking e large enough so that d = 1 is instructive as well. As long as the  graph enters and leaves through the sides of the box everything is wonderful.

 

Of course the box is the same box that we've all drawn on the board and is in all the textbooks. But the dynamic role played by the sliders is far more instructive than the dynamic hand waving with a static picture.

 

Hope this helps

Grading

Let's think about grades for a minute. The percentage of the total points on a test, quiz, project, etc. seems to be what everyone thinks they should give their students for a grade. But this carries a lot of assumptions, which IMHO make little or no sense. Percentage grading assumes:

• You can write a test or judge another type evaluation so that a grade of ,say, 80% means the same thing each and every time, and that the minimum "passing" grade is that much better than one percent less than that grade.
• Each test has the same difficulty, so that 80% always means the same thing through the year.
• That the teacher across the hall teaching the same course, will give percentages that mean the same thing as yours, and the teacher in the next school district, the next county, the next state ...

Then what happens? All your grades for a particular test are low. Does this mean the test was too difficult or that the students did not learn all they should? What to do? Some folks "scale" or "curve" the these tests (but not the ones where everyone does well), or they make sure the next test is "easy." Then of course you have to justify scaling only the "hard" tests and not the ones where everyone does well. (This in fact assumes that the test was too hard -- else why reward kids who didn't learn what they should have -- and proves you can't judge the difficulty of your own tests.)

Percentage grading only seems to work for very experienced teachers who use the same tests from year to year and have adjusted them so the grades come out the way they would like.

There are various ways to scale or curve tests.

• One that has been discussed recently is the "square root curve" where the grade is 10 times the square root of the percentage. This raises everyone's grade; those in the middle a lot, those on either end a little. (Max/min question: which percentage gets raised the most? Which the least?). I can see even less justification for this approach -- except to raise everyone's grade.
• Other schemes are to list the grades in order and look for natural breaks. Highest group gets A or 90s, next get B or 80 etc. Somewhat arbitrary, but may make more sense than anything above.
• Find the z-scores and force the grades into a normal distribution -- this assumes your kids are normally distributed and in an AP class that may not be the case. Besides with this approach you will always have to fails some one even if they are all pretty good students.
• Throw the test up the stairs ...

So what to do? After all you do have to assign some grade. My suggestion is to read "Assessing True Academic Success" by Dan Kennedy. A perfect approach, probably not. But it will give you some things to think about and suggest a curving scheme that you might like and that has some thinking behind it. 

The take-away from this article is "You control the grading algorithm."

The Shell Game

It's been a while since I've written anything here. The usual excuse: too busy with work. Here is a little idea you may like. 

Every year about this time as AP Calculus teachers get ready to teach applications of the definite integral the question comes up about whether to teach the method of cylindrical shells. There are several things to consider.

First, the method is not tested on either AP calculus exam; so you will not put your student at a disadvantage if you omit the topic. Any volume of rotation question on the exam can be done with the disk/washer method. But then it is technique that all the texts cover and most college teachers expect students to know. So if there is time it is worth including. It is probably best to include it when you do other volume problems. Due to time considerations some teachers save the topic until after the exam, during the final weeks of school.

However, what I'd like to discuss here is the fact that you never actually need the method of cylindrical shells. You can always avoid it. That said, let me add that shells is usually faster and more efficient than the idea that follows. The example shows how to work around the shell method (pun).

This technique will work for a function that is one-to-one on the interval of integration. If your function is not one-to-one, the interval must be broken into sub-intervals on which the function is one-to-one.

Here is an example: Find the region enclosed by the graph of the lines x = 1,  y = 1, the y-axis and the graph of

is revolved around the y-axis to form a solid figure. Find the volume of this figure.

Proceding by the disk/washer method the volume is given by

 

Note the dy and the limits of integration that correspond to the dy.

The usual next step is to write the integrand in terms of y so that the antiderivative can be computed. Cubic equations can be solved, but who remembers how? And even so the next example may involve a sixth degree polynomial, so let's assume you cannot solve for x in terms of y.  At this point textbooks give up and recommend the method of cylindrical shells.

But it's not necessary that the integrand be in terms of y, it is only necessary that the integrand contain only one variable.  So consider the substitution:

Making this substitution we get

 

The resulting integrand is a polynomial that can be easily expanded and antidifferentiated. Note the change of the limits of integration to correspond with the dx.

Do AP students need to know this? No. It is more of a curiosity or a little "extra" for them. But it is a nice idea and also illustrates a slightly different substitution.

Using CAS for testing

The ETS recently posed a survey concerning the use of a symbolic manipulation calculator on their exams. Here are my thoughts on the survey.

 

First, the survey seemed to be aimed at determining whether a symbolic manipulation calculator (SMC) would give someone with a SMC an advantage on the standard questions that appear on the ETS exams (AP, SAT etc.). The questions did not seem at all the kind were designed for SMC use. Too bad.

 

The argument is always along the line of if using a graphing calculator (GC) or a SMC helps people learn more or better or deeper. That's different than what should be expected or required on an exam of one's own or of the ETS. So I'd like to discuss two questions to see the difference between good testing practice and using SMCs to do and learn math.

 

2003 AB 76 gave a velocity function v(t) = 3+ 4.1cos(0.9t) and asked for the acceleration at t = 4. The point of the questions was to see if the student know the relationship between velocity and acceleration. The problem could be done be hand; this requires the use of the derivative of a constant rule, derivative of a sum rule, the constant multiple rule, the derivative of the cosine and the chain rule. To do this by hand would necessitate the use of "easier" numbers and one of the special trig values. A student who makes a mistake on any of these gets the wrong answer and leaves the examiner unclear as to whether the student knows that the derivative of velocity is  acceleration. Therefore without the use of a GC the question is pointless.

 

But you say all those other things are important; a calculus student should know them. I agree -- but they were tested elsewhere on the same test. The velocity - acceleration thing was the point here.

 

2003 AB 87 gave f ' (x) = sqrt(x)/(1 + x + x^2) and asked for the x-coordinate of the point of inflection. The fastest and, I am sure, the intended method of solution is the use the GC to graph the given derivative and identify its extreme value which corresponds to a point of inflection. This concept is often tested and can be found on several free-response tests.    

 

The questions could be done by graphing f '(x) by hand, but even with a simpler function with easier arithmetic this is time consuming and not practical. You could also find the second derivative and find its zeros by hand. Again not really practical with the given function. It is correct approach but one that does not tell us that the student knows the graphical relationship between the graph of derivative and the POI.

 

Using a SMC you can more easily find the second derivative and its zeros. So you can get the answer, but even that takes more time and thought than the intended method. So here the SMC will work but is a disadvantage time wise. And, as before, does not tell the examiner if the student knows the relationship between the extreme values of the first derivative and the POI.

 

The point is that technology can make testing more focused on what you want to test. Using more complicated functions and less "easy" numbers is a way of pointing the student to his GC or SMC hoping he will show that he understands the concept being tested.

 

Whether you choose to let students use a GC or SMC on your tests is up to you; you can certainly test these and other concepts without them. Certainly you can write questions for which a SMC will not be of any help. Or you can use the SMC to focus on the concept you are interested in. I think that's the point in testing with a SMC.

 

But the other reason to use a SMC or other technology in your class and requiring your students to know how to use it is to help them learn math better. I could live with no technology on test, but not with no technology in class. The reason to require technology use on tests is to be sure it is used in class.

Technology Changes Curriculum - Or does it?

Technology changes things. That's sort of an understatement.

The first big curriculum changes resulting from technology occurred some 40 years ago. Four function calculators became available. I remember my first one cost $70. At that time something between 2 weeks and a month was spent in Algebra 2 classes teaching computing with logarithms.

For you young folk: back in "the old days" calculations involving multiplying, dividing, power, and roots we routinely done using logarithms. The logarithms were listed in tables. A  logarithm table for 3-digit numbers is two pages long, 4-digit numbers 10 pages and 5-digit numbers 100 pages. While in college I worked summers for a civil engineer; he had an even larger table that took up an entire oversize book -- and he used it. There were also tables of the logarithms of the trigonometric functions.

The were no calculators. We had in the office a machine to do multiplication and division. Huge thing; rows and rows of buttons. You entered one number and then to multiply you turned a crank for the other number one digit at a time. For the next digit you turned another crank which move the whole carrage over one place so you could crank in the next number. To divide you turned the crank backwards. Hence the term "crank out the answer." Someone good with logarithms could do the job just about as fast.

The other thing close to a calculator was the slide rule. But the slide rule is just an analog representation of logarithms. 

Students in Algebra 2 were taught all the tricks of calculating with logarithms: how to deal with the decimal point, how to interpolate between the values in the table to get more accurate answers, how to use co-logs to avoid subtracting and of course how to use the table in the other direction to change  the final answer back to a regular number (called finding the antilog).

Multiplication and division were done by adding and subtracting the logarithms; powers and roots by multiplying or dividing by a  single-digit number. I remember after teaching how to set up the problem and find the logarithms that one year the students did all the set up work, looked up the logarithms and then got out their calculator to do the adding and subtracting! That's when I know logarithms were on their way out.

There are lots of topics in mathematics that were invented to make calculations with numbers easier and more efficient. Logarithms was certainly one of them, but there are others. The advantages of four-function calculators and soon thereafter scientific calculators were obvious.

But, here's the thing, computer algebra systems, CAS, do not seem to have had the same effect on the curriculum. Why? There are many basic computations that we still require students, not just to know, but to be proficient at doing by hand. I'll list a few below, but so that I don't sound too far out, let me first say this:

Just as logarithms are and always will be important and students need to understand them, so too the things in the list are important and students need to understand them. But they don't have to be good at doing them! Just as very, very people who graduated high school in the last 30 odd years could come close to doing a computation using logarithms, students in the future don't need a by-hand proficiency with these thing. They need to understand what's going on; they do not need to be good at doing it.

Okay, here's my short list:

• Simplify radicals
  
• Rationalizing denominators (or numerators for that matter)
  
• Factorig quadratic polynomials (those few that actually factor)
• The Quadratic Formula and the "discriminant" (although completing the square is a keeper, but not for solving equations)
• Cotangent, secant and cosecant (these have not been taught in Europe for decades, if ever. Just noticed that the spell check on this Blog doesn't recognize "cosecant" as a word; maybe there is hope.)
  
• Newton's Method for approximating roots of any equation
• Antidifferentiating (there are tables of over 1,000 antiderivatives in print, these too can go)
• Synthetic division
• Descartes Rule of Signs
• The Law of Sines and the Law of Cosines
• Cramer's rule and the discriminant
  
• Matrix arithmetic and find the inverse of a matrix

 So that's my short list. Anyone care to add anything?

They were invented for a good reason; but that reason no longer applies.

Why hasn't technology changed these things? I think the major reason is that they appear on the state graduation exams -- now there's a good reason to teach them!

CAS question

As I have made no secret of the fact that I think symbolic manipulation software should have a major role in mathematics education starting no later than Algebra 1, I tend to get into a lot of discussions with folks who do not quite agree with me.

Not too many years ago graphing calculators made their appearance in high school math classes. They were pushed down from the top, so to speak. By requiring students to have them for the 1995 AB and BC calculus Advance Placement exams, teachers were forced into using them. What happened? Almost overnight those teachers and their colleagues saw how much more students could learn and understand using graphing calculators. I doubt very many would give them back today.

The very first models graphed and did numerical computations. Very quickly the ability to make tables of values of functions and software to do sophisticated statistical calculations were added.

(Side track: That first year, 1995, I used the TI-85 which did not have a table feature and half my class showed up with HP-48s (don't ask) which did not have a table feature either. What to do? I wrote a program for the TI-85 to make tables and one of  my students wrote a table program for the HP-48 (in assembly language no less). We shared them and then everyone had a table program)

Today, computer algebra systems, CAS, include those features and also include the ability to do symbolic manipulations. (And more actually: dynamic geometry software and computer like document handling are features of the newest models.) While CAS do all the graphing and numerical work that graphing calculators do, the acronym CAS has come to refer to the symbolic manipulation features alone.

So here is my question: Since the graphing, table, statistical, and geometry features of the CAS have been embraced, why not also symbolic manipulation features?

Any ideas?

Making the Grade - 3

So if percentage grading doesn't make it and the various scaling schemes have little to recommend them, what do I recommend?

Before I answer that I suggest you read ASSESSING TRUE ACADEMIC SUCCESS: THE NEXT FRONTIER OF REFORM by Dan Kennedy of the Baylor School in Chattanooga, Tennessee. This article deals with a lot more than scaling test and is well worth reading. Dr. Kennedy is the co-author of a popular calculus textbook and has been involved with the AP Calculus program for many years. His wide ranging ideas on assessment make a lot of sense (to me at least). 

As I discussed previously, scaling is often used to raise students' grades when the teacher has given a test that is too hard, when the mean score was too low. Dr. Kennedy writes, "If you want your students to think on the test, then you will have to give them a question for which they have not been fully prepared." Certainly posing questions for which the class is not fully prepared will make the test "too hard." His answer is to stretch the students by asking challenging questions on the tests and the protecting then students' grades by scaling the results. He points out that "[you] control... the grading algorithm." His conclusion is that "we ought to present students with challenging, relevant, useful, and varied assessments all of the time, and then scale the grades to conform to our expectations." (Emphasis in the original)


Here is Dr. Kennedy's suggestion on a method for scaling test grades. Take two order pairs. The first is (class mean on this test, desired class mean) and the second (the highest test score, 99). Write the equation of the line containing these two points and use it to scale the grades. (The "desired class mean" can be set in advance and adjusted slightly and can be different for different level classes. It can be the historic mean for each course.)

He finds that,

Freed from the shackles of unreasonable numbers, I can now challenge my students to do just about anything, then see how far they can go. They, in turn, have been freed from the burden of getting a certain percentage right, so they can concentrate on doing as much as they can as well as they can.

What more could you ask for? 


Again, I suggest you read  the entire article. It suggests other methods of assessment that the author has used in mathematics classes.


While no scaling method is perfect and will work for everyone, I think that this one has a lot to recommend it. First and foremost you can and will ask challenging questions. You also will be able to test that do not total 100 points. If your test is too long for the time allowed, well it's too long for everyone and, because of  the scale, no one will be hurt.