This week in my precalculus class I used two worksheets that seemed to engage my students more than other worksheets have in the past. The topic was piecewise functions, functions formed by partitioning a domain into subdomains, or pieces, each of which piece is then associated with a dedicated function. For example, for a hypothetical parking lot, charges for the first hour could be $10 dollars, the second hour $8, and every hour after that $5, up to 24 hours (with partial hours rounded to the next hour). The function to calculate total charge for parking could be represented as follows,

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
f(4.5) = 10+8+5(5-2) = $33

The piecewise function has two parts, the right-most part that specifies qualifying conditions and the left-most part that specifies a dedicated function to evaluate, if its associated condition is met. That is, the unique condition in the right-hand part, that evaluates to true for the input to the function, identifies its corresponding dedicated function in the left-hand 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 TI-84 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 in-class 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 age-based 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 in-class 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 in-class 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.


Bits and Pieces

This week, rather than reflect on a topic in teaching, I’m just going to list bits and pieces of my teaching environment that I someday need to think about, do about, or forget about.

  1. I am teaching introductory calculus this quarter, the first section of four, and the focus is on differentiation. While wandering the internet I came across an online differentiation calculator that can easily differentiate all of the standard equations that I generally assign for homework practice. It will even show the steps that were followed. So it begs the question, why assign equations for students to differentiate for homework? And why ask them to show their work?
  2. I generally write my lesson plans, quizzes, tests, worksheets, etc., in MSWord using its mathematical editing application. It has worked very well for me in the past; I’ve been using it for years. Last week, however, MSWord began to inexplicably slow down after I was editing one of my files with a lot of tables with embedded graphs, text, and mathematical equations. I would type and nothing would happen on the screen. Some 10s of seconds later, my typing would begin to display.  Needless to say this is not an acceptable way to work. I ran some searches to see if anyone else had the problem and might suggest a solution. I found some references to Windows 7 and Office 2013 showing this problem, but the references were several years old. I read through some of the suggestions, and tried a few that did not require editing the registry, but no luck. Editing still seems to slow down at unpredictable times on larger files with lots of tables. I then had the school’s help desk look at the problem, but the best they could suggest was that I clean up my trash bin and temporary files and defragment the disk. None of that did anything to fix the problem. So I’ll have to think more about this next week. Oddly enough, I have Windows 7 and MSWord on my home computer, but the same files do not create the same problem.
  3. I’ve been struggling with testing and grading this quarter. It just seems to take too much time to prepare weekly quizzes or tests, including answer sheets, and then grade the weekly quizzes or tests for 100 students. I want to offer weekly assessments to keep the students engaged, but it’s really becoming too much work for me. I have been avoiding multiple choice questions, but I may be forced to try them. However, it is always more work to change methods in mid-stream so I’ll wait until next quarter to begin the change. Of course I’ve been warned that the preparation of good multi-choice tests can be as time consuming as the preparation of any other type of assessment.
  4. This week I had trouble following the work of a student on a quiz. The flow of mathematical argument seemed to make sense, and then, suddenly, it didn’t. It takes concentration and time to try to guess why a student suddenly drops exponents in a differentiation when there is no logical reason for dropping them. Some rules of differentiation can have that effect, but the context didn’t call for the rules. I stopped and wrote a note to the student to go through my online solution to the quiz and then get back to me with questions. Maybe he can figure out what he’s doing, or not doing.
  5. No matter how much I discourage the question, students just can’t seem to resist: “Is this going to be on the quiz?” Maybe I should take those questions as a message to me about my teaching, my lesson structure, my rules of engagement. Or maybe I should always just answer “yes.”


A Question about a Question

In a recent email a colleague wrote the sentence, “The ability to read simple directions seems to be a dying art among students.” The context made it clear that he was jesting, but I was struck by the sentence because I had just finished grading a quiz in which a majority of my students had misread what I considered a simple direction. The question that was misread was the following:

  1. (6 points) If you invested $5,500 in a banking account, what is the final balance in the account and the amount of interest paid after 4 years if you earn:

1a. 1.7% interest compounded annually?

1b. 0.7% interest compounded continuously?

For this question, I expected the students to calculate two dollar amounts for each of parts 1a and 1b. One dollar amount giving the balance in the account after 4 years and one dollar amount giving the interest earned in the account after 4 years. Of the 32 students who took the quiz, only eight answered with both the balance and the interest earned while 24 students answered with only the balance.

Although it is tempting for me to dismiss the result as students moving too quickly through the quiz, I am afraid that there must be more to this than just the students’ direction-reading abilities. After all, more than 70% of the students missed the cue. There must be something about the question that is broken. My current thought is that students forgot about the interest part of the question because what they were accustomed to calculating in class was the final balance. We may have looked at interest apart from balance a few times, but the majority of practice was in calculating final balance. I presume they mentally closed the question once they found the balance, and moved on.

If that, possibly, is what happened, how should I rephrase the question to make both parts of the question more memorable after the balance calculation? Should I provide fill-in prompts for each of parts 1a and 1b, one prompt for final balance and another for interest earned? Should I explicitly rewrite the question to emphasize its two parts? For example,

  1. (6 points) If you invested $5,500 in a banking account, what is the final balance of the account and what is the final amount of interest paid into the account after 4 years, if you earn:

1a. 1.7% interest compounded annually?

1b. 0.7% interest compounded continuously?

Or should I rewrite the one multi-layered question as multiple different questions? Or should I provide a fill-in-the-blanks answer table. Or is there something else I should do? More work for me, but the question about the question needs to be answered.

And one more thought. It was not a good idea to use 1.7% in part 1a and 0.7% in part 1b. A few students answered both questions using 1.7%. I’m sure it’s one of those tricks the brain plays when in fills in what it expects to see, rather than what is written.

Until next week.

Lessons on Lesson Planning

Today I’d like to reflect on lesson structure and lesson preparation. In the past I’ve prepared for a lecture by writing lecture notes on 8 ½ x 11 notepaper. The notes would be structured to include, in order, all the topics I wanted to cover that lecture period. The notes would also include examples for me to develop at the board and a short exercise for students to work on after I completed my examples. When students finished the exercise I would poll them for answers—usually there were 2-3 different answers called out. Once all the answers were taken I would walk (figuratively) through the exercises with the students, as a class, noting the places where wrong answers were creeping in. Then, if there were no questions, on to the next topic.

As I’ve written in an earlier blog, I’ve being trying to change my lesson structure to make my lessons more active. I have been including worksheets in my lesson structure, with more complicated questions on the worksheets, in place of the simple in-class exercises I was previously using. The worksheets have about 15-20 minutes of material in them and they normally require a student to interpret English-language descriptions of problems, rather than just mathematical formulas. As the students work on the material, alone or in groups, their choice, I circulate and answer questions. So far, the change is helping me to work more directly with students who are having trouble with the questions—either the math or the interpretation of the English description. These students did not self-identify as having problems when I was using the older structure of embedded, single-exercise check-ups on understanding. I’m positive on the changes so far. Unfortunately, I haven’t been doing this long enough to say much more about it right now.

Another change I’ve started is to enter my notes in outline form in computer files. The advantage is that I can hope to create an improving body of lesson plans as I teach and reteach the same courses. The disadvantage is the time it takes to format the lesson plans online—I’m accustomed to thinking with paper and pen. It’s also time consuming to generate graphs and tables that are simple to do by hand, but not always so simple in a word processor. Some instructors scan in hand-draw diagrams to get the best of both worlds, but I haven’t tried to do that yet.

I have, from the beginning, created my worksheets online. It’s also a time consuming process, especially since I create an answer sheet to share with the students. (Unless I just run out of time. Then the students have to be satisfied with in-class answers.)  I do like having the growing body of worksheets, but I now need to give more thought to how I name the files, since I’ve already experienced retrieval problems and I’ve only just started with this method.

For now, I have to return to lesson planning. More next week.