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.

Too Tired Tonight

I’m too tired tonight to reflect on my week as an instructor of mathematics. It was one of those weeks that was consumed by lesson planning, teaching, grading, office hours and meetings. What’s that you say? Why was this week any different than others? It wasn’t and somehow it was. I do all of those things, every week, but this week I tripped over the professional/personal life-balance line. I had a ‘mid-term’ to grade for a calculus class and two quizzes to grade for two pre-calculus classes. And a recommendation letter to write for a student. And meetings on topics of various importance. And lesson preparation for five days of classes. And homework to grade for all three classes. Not a lot of homework. Most of the homework I assign is on computer-assisted systems, but I feel it necessary to grade a few problems each week that are more challenging—some practical and some theoretical. They are on paper. They add up. They end up at home. They need to be sorted. They need to be graded. That can’t be done with too much distraction. Four problems each from 100 students. They take time to read and interpret (the problems, not the students—or maybe both). There is a spectrum of styles of written homework, in multiple dimensions: from obsessively neat to hopelessly tangled; from verbose and repetitive to sparse to the point of no-credit; from smudged pencil to calligraphy; from whatever to whatever. The results need entry into spreadsheets. And the spreadsheets need to be synchronized (light beams from the cloud touching here and there with varied filtering). And the averages need to be noted—too low or too high? And what about next time? Did the students learn? Did I learn? What should I change? Are the answers still mine or running free? But, enough is enough. I’m too tired tonight to reflect on my week as an instructor of mathematics.

The Easy Question at the End

I gave a test this week in my pre-calculus class and I’d like to reflect on one test question, the last one of seven. The question set up a realistic problem that modeled the measurement of light intensity using a rational function, the topic of last week’s study. The rational function modeled the intensity, in lux, that a measuring device would detect when placed meters from a low-intensity bulb and, correspondingly, meters from a high-intensity bulb—the bulbs were at a fixed distance apart—ten meters. The details of the function are not important, what I’d like to reflect on was the students’ response to the question.

The question had two parts:

  • Identify the practical domain of the rational function for this application.
  • Use your calculator to estimate the minimum and maximum distances from the low-intensity bulb that would register less than 5 lux on the measuring device.

First some background. The week before the test I presented, at the board, as a lecture, a very similar problem. So similar that it showed two light bulbs ten meters apart and a measuring device between them that was modeled by a very similar rational function. We (at least a few students and I) discussed the practical domain of the rational function, given the setup; how it only involved positive values, and in particular, values between zero and 10 meters. I thought the class understood. I then went through an analytic solution of the rational function, which eventually involved the solution of a quartic equation. The solution was long and messy (it was supposed to be). After that, we solved, as a class, on our calculators, the same problem by using a graph of the rational function and a graph of a horizontal line at to find the two points of intersection of the two graphs that would answer the second question of the problem (for a value of 4 lux). I displayed my graphs, in the practical domain, on a screen at the front of the room, and the students, supposedly, created the same graphs, and found the same points of intersection, on their own calculators. The day before we had gone through a different problem that was difficult to solve using a graphing calculator, but straightforward to solve analytically. The two problems were meant to stand as endpoints on a spectrum of problems that were variously easier or harder to solve analytically or graphically.

So how did my students do with a problem that I thought was a gift (and therefore assigned fewer assessment points)? Here’s a summary table of the results.

Total Answers Correct Answer Incorrect Answer
Identify Practical Domain 34 15 19
Analytic Solution Attempted 11 11
Calculator Solution Attempted 11 5
No Solution Attempted 7 7


I think I now understand why a well-formed lecture and a seemingly attentive class do not necessarily mean that the students are learning what I think I am teaching.


Talk, Walk, Repeat

I’m an instructor at a community college in California and I teach mathematics. This quarter I’m teaching pre-calculus and calculus. As a part of a professional development program at the college, I have committed to reflecting on and writing about my teaching experience, once a week, for ten weeks. Here is my first reflection.

This quarter I’m also trying to use more active-learning techniques in my classroom. I’ve taught, up to now, in a conversational-lecture style, but comments from my colleagues and students during the last quarter have convinced me that I need to teach in a more student-engaging way—one that requires more active and less passive learning from the students. For right now, taking change slowly, I’ve altered the structure of my lesson plans to the following format: lecture, worksheet, lecture, worksheet, etc. The idea is to:

  • Lecture on a topic for 15-20 minutes.
  • Let students try to solve topical problems, alone or in groups, for 10-15 minutes.
    • During which time I wander the classroom to observe and converse.
  • Let students present solutions, with open conversation, for 5-10 minutes.

Not very original, but it gives me a way to start that is active-learning and lecture together—just a little outside my comfort zone.

This is week two of the quarter. Are my changed lesson plans working? Are my students more engaged? I am not sure. I need to find a way to evaluate what I’m changing, but since I’m in the middle of the change it’s hard to stand aside and just watch. And what should I be watching for?   I have seen concentration during problem-solving by students, and I have seen them working in groups of two-three, but I have also seen play-working by individuals and groups. None the less, without any way to justify it, I have an impression that the students are more engaged by the insertion of the activities in place of straight lecture. (And why shouldn’t they be? Who can concentrate on listening for more than 20 minutes?)

I’ll continue the structure next week and try to improve the continuum of challenge in the problem sessions. I’d like to engage all the students so none are too bored or too challenged. But, that too, I’m not sure how to measure.

Time to walk.




Without Accountability There is Often a Gap

My last post was months ago, after the first week of the quarter.  This post, only my second of the quarter is occurring immediately after grading the final exams.  I don’t have a good excuse for not blogging; no more than my students have a good excuse for not studying.

This was a challenging quarter.  Many of my students were not interested in the material, or in doing the work, or they just found the work too challenging.  And of course there were the 1% who were bored, learned nothing and did well on the exams and homework, if they bothered to do the homework.  Some of the students spent their time on other courses that were more important to their plans, calculus, or chemistry, or biology, or …  Even many of the computer science majors seemed less than interested in discrete mathematics.  For this class I should have had more in-class assignments.

The final exam was a special disappointment.  Apparently, many of the students did not study for the exam.  There was a lot of material and I did not give them a practice final, so they had to be self-motivated.  Only about 10% of the students received an A grade on the exam.  Most would have failed if I had not graded it with generous credit for work that went wrong.

I wish had been able to get to know some of the students better.  I find it hard to remember all the names and to just make small talk when I do see them out of class.  Next quarter?

Can there be a difference?

A new quarter has started.  I have already taught five class sessions of discrete math.  I’m following the same format as last quarter with only slight changes to the assigned homework problems.  There was only one week off between quarters so there was no time for substantial change in the form or content of the course.  Some changes are pending but they will have to wait until summer break and the fall quarter.

Some notes on the first week.

  • I have more students this quarter who are having trouble following the lectures and completing the homework.  A few have approached me after class to say that they don’t understand the homework or that the homework is too hard.  A few have dropped the course.  This phenomena of early drops and start-up anxiety is not unusual, but this quarter it seems to be more pronounced.  Normal variation in class behavior?  A different profile of students?  I’m not sure.
  • One student, a non-native speaker, approached me after class to tell me that she could not understand the text book.  She read the chapter and used a dictionary to try and understand some of the mathematical terms that she did not recognize, but even with the use of an interpretation dictionary the meaning was not clear to her.  Discrete starts with a development of propositional and predicate logic.  Both require a focused look at what lies beneath informal mathematical so that the informality can be formalized.  I know that it is tempting to just present the mechanics of logic and stay away from the translation of English-language statements into and out of the mechanics, but the students need to ‘stand under’ the formality, not only ‘stand on’ the formality, so that they come to ‘understand’ mathematics.
  • I gave a quiz on the second day of class.  A quiz on an introductory chapter that the students were assigned to read on the first day of class.  The questions on the quiz were mostly about prerequisite knowledge so, in theory, any student should have been able to take the quiz without reading or studying.  Some were clearly ready.  Some were clearly not ready.  I did something with this quiz that I’ve never done before.  I graded the quiz without deducting points; everyone who took the quiz got full credit, but it was obvious from my comments and corrections on the quiz when a student had not earned it.  I did this both for them and for me.  The first week of class is chaotic; some students would not find the time to prepare.  Some students added the class on the day of the quiz.  Why discourage students with a harsh assessment on day two?  But I need an idea of what the students know, or need to work toward.  Right now I am satisfied with my approach.
  • Community colleges are reemphasizing  equity again, trying to keep more students in the game longer, with more successful outcomes.  How does this translate to teaching?   The curriculum doesn’t change; the course outlines don’t change; the time in a quarter does not change.  What is changing?  Tutoring and consoling resources are increasing and early recognition of problems is being encouraged.  I’m trying, this quarter, to identify early students that seem on the edge in any way.  Sometimes it’s easy–the students self identify.  Sometimes its hard–there are always students who start out strong and finish weak so it’s harder to know of their troubles until it’s late in the quarter when it’s more difficult to do much about troubles.  But this quarter it’s my goal to not make excuses for myself, but rather to identify the students who need extra help, early.

That’s it for this week.  We’ll see if things are different as the quarter unfolds.


The course has run its course. The final is finished.  The red ink has dried.  The assessments are formed and recorded. Did everything go as expected?  No.

  • I thought I edited the test to fit its demands to two hours, comfortably, for any prepared student; I was wrong. Approximately 1/3 of the students needed extra time, and many were my well-prepared students.
  • I thought all my students understood the cardinal points of cardinality of sets; I was wrong.  Many students completely missed the finite/infinite distinctions.
  • I thought my students understood the subtleties of counting team formations; I was mostly wrong.  Many clearly had never read the examples in the text.
  • I thought my students understood a notation I introduced early in the quarter and used almost weekly; I was mostly right, but some students had no idea what I was asking.
  • I thought I had proof-read and edited the test to the point of clarity of all questions; I was wrong on at least three of 15 questions.

What about the architecture of the test?  Did it meet my needs?  Here I can be positive.  The test was divided into three sections.  The first section tested basic understanding of definitions and methods.  The second section integrated concepts but sought definite answers.  The third section was open-ended and devoted to proof.  Having the sections so divided did give me a profile of understanding for each student.  If the test had been a little shorter the students may have had time to polish their proofs at the end; but shorter tests magnify errors.  Overall I was pleased with the form and function, even with some very rough edges.  I’ll use the same blueprint again.  But now it is time, finally, to rest.

The Week Before Finals

The quarter is almost complete.  I presented my last lecture on Friday–it dealt with the concept of a function being the ‘order of’ another function.  When developing algorithms it is often tempting to accept the first draft of software that appears to implement the algorithm faithfully–it produces the expected answers for specified input.  But that’s never sufficient for commercial use.  Early software may work very well for an average demand, but may fail completely in stressful situations.  For example, too many simultaneous requests for service may drive software into an infinite descent of queuing with neither time nor memory left for serving the queue.  One way to test for how explosive a demand on resources may become is to model the order of the most-used algorithm that underlies the software.   To model the algorithm one has to have a function, let’s call if f, that reflects the use of resources based on an independent variable such as service requests per second or size of the data to be processed.  Developing such a function is artful work.  When function f is available then it needs to be confined in its behavior by a simpler mathematical function, let’s call it g, such as a single-term polynomial function, or an exponential function.  By ‘confined in its behavior’ I mean f must be bounded by g.  That is, there must exist positive constants A and B such that A|g(x)| <= |f(x)| <= B|g(x)| for x larger than some non-negative constant k.  But, I’m straying too far into the subject when I am supposed to be reflecting on the teaching.

The final is all that is left now.  The students are nervous about it but I think they are ready.  Writing the final took more time and thought than I first expected.  Like all tests, it has an architecture–a form and function that work to satisfy the needs of a client.  In this case the client is me and my need is to assess what my students have learned through the quarter compared to what I planned for them to learn.  After several false starts, I settled on the following form:

  • Two pages of comprehensive basic-knowledge questions
  • Four pages of integrated-knowledge questions, including application-related questions
  • Three pages of formal proofs with the students able to select two of five proofs to complete for grading

The function of the exam was set from the beginning,

  • test for basic understanding of definitions and methods
  • test for integration of basic understanding
  • test for facility with mechanisms and applications of formal proof methods

I expect all the students will be able to complete the test in two hours–if they are prepared.  The last time I taught this course the students, prepared or not, were not able to complete all of the questions.  I hope that I have written a better test this quarter.  I’ll know next week, the week after finals.