When Descriptive Becomes Prescriptive

When the wisdom from years of teaching interfere with your teaching.

After teaching the same course several times over the years it is natural to develop some ideas about how it typically goes for different types of students. For example, in a course that enrolls both mathematicians and engineers it is not unusual to find that there are parts of the course that one group does better with than the others. Whether or not it makes sense or is appropriate to adjust the course to account for this difference, is not the focus of this post, but I’m more interested in what becoming aware of this difference and possibly letting the class know of it does to the course. My basic conclusion is that unless you spend a lot of time and energy on resolving the issue, saying anything to the class about it, or even dwelling on it too much yourself, will only cause problems.

Suppose in the above case you announce that ‘Engineers don’t do so well in this part of the course.’ First we have to realize that students usually hear statements from the professor as proclamations of truth; but that it is usually truth by position, i.e. it is true because the professor will make it so. So upon hearing the announcement it is likely that some engineers will interpret it to mean that part of the course is designed to not be successful for them. Even if the students don’t think it has to happen, we then get to the question of who should do what about it. If the announcement is made, the usual follow up is something about ‘working harder’, which implies that it is up to the student.  The student may hear the announcement and think either that it is out of his or her control, or that the professor, knowing that it is a problem, will do something to make it okay. In either case, the result is no extra effort by the student. For the students that the announcement doesn’t apply to, they also may hear some other messages, like that it will be successful for them or that it will make more sense to them naturally (a question of impact on grades or learning). They may also think that they will do better as there is a group of students who are already designated as doing worse (via a curve). The problems start when the expected adaptations don’t happen (or at least at the level that was expected), or even when the pronounced results do happen. This ties into the typical student response to unexpected failure: ‘Why did or didn’t the professor do so-and-so?’  But now it has new legs as there is some evidence, via the announcement, that the professor knew something was going on and either didn’t do anything to fix it, or did things to make it worse (or to actually happen). So the announcement that was probably intended to cause heightened diligence (and glory to the professor for pointing it out to students so that they could be successful) actually turns into less effort and possibly very negative responses.

But what if you don’t say anything, but you still are aware and are thinking about this difference? Suppose you choose not to do anything about it. Then you still potentially have the same results as the students in the different groups won’t know that they are more or less likely to be successful and won’t make any adjustments, then the performance on the assessments will be poor (for some), and you won’t feel the need to do anything because you already expected it. So again you are caught not doing something to help, and, if pushed, would say that you expected it and so aren’t going to do anything about it. Suppose instead you do something about it (without telling the class why). So either you adapt the course so that the difficult material is avoided (for all or some), or change the presentation in a way that favors the previously unfavored group, or you change the way you grade to lessen the difference. So you either lessened the course or you’ve introduced a bias. In both scenarios, another consequence of even being aware of such a difference, is that it is probably wrong and will cause some individually unfair differences in the class. Like other –isms, if we assign some characteristic to a group (true or not), we end up treating them differently and not treating the individuals as individuals.

What if the observation and description applies to the whole class? For example, through experience, you know that students find some part of the course to be much easier than another part. I think you still have the same issues as above whether or not you share it with the class. Anything you say or do about how the learning could/should/will go in your class, leads to contrarily indicated responses by the students or negative changes in students attitude or behaviors towards your class. For example if you say something is hard, you have some students that expect you to adjust for that, or others who have struggled before that will more easily accept failure. You might skip the material, water it down, or just grade easier. You can have strange mixtures where you say it is hard and increase your effort to explain it, and then have the students think it is easy as they never came face to face with the difficulty of the topic, and then your proclamations are devalued or the students get an inflated sense of their knowledge.

It seems we’ve come to a point where any sort of learning observation becomes a potential negative no matter what you do with it. Yet, those sort of observations are at the root of most teacher development: I’m trying to have the students learn X, via experience Y, and I then check to see how we did, with any observations that learning didn’t happen overall or within certain groups, leading to modifications of X and/or Y. I think we still need to make those observations, but need to be more careful in how we use them.  Maybe the best plan is to keep the current design of the course and let those observations modify the next iteration of the course.


Books are Good, Thrift Books are the Best

A few random ideas about and from books.


In Praise of (Thrift) Books: I like reading books, or, if the stack of 6+ books at my bedside is any indicator, I like starting to read books. My favorite sources for books are the local thrift stores because they are cheap and you never know what you might find. The result of this combination is that I walk out with some interesting books I would never buy for full (or even Amazon discounted) price. A hardback with an interesting title and/or back blurb for $0.50 is an easy buy, and if it doesn’t turn out to be worth the time to read, I don’t feel bad donating it, giving it away, or trading it at a used bookstore.


Random Thoughts from the Current Set of Books: I believe that you can learn something from every book, even if it’s something like “Don’t ever write anything like this ever!”, and if you are looking for ideas related to teaching, you can find lots of great nuggets in just about every book.  So here’s some teaching ideas I’ve garnered from 3 of my bedside crop of thrift books:


Cover of "Good to Great: Why Some Compani...
Cover via Amazon


Good To Great by Jim Collins:  As the title blurb says, it’s a study of why some companies do well and others don’t.


– If you want to have high quality results that you can share with others, you have to invest heavily in research and document it all carefully. The research in the book is supported by a team of researchers and every idea is carefully researched and footnoted. If the book had just been one person’s ‘good ideas’, it wouldn’t have much impact. Our efforts at improving our teaching should be treated similarly.


– Success or failure should be measured against the changes that are distinctive. It is easy to look at some good (or bad) teaching and associate it with all the apparent behaviors, but it is more helpful to look at what practices are different with the teacher and students and connect it to the good (or bad). In the book, they compared successful companies and practices to those that were similarly successful up to  point.


– Ideas matter more than people. In the book, they noted that ‘celebrity’ CEOs might have initial impact in a company, but for sustained improvement, CEOs that had and followed a set of success-focused ideas worked out much better. We shouldn’t just judge teaching by the star-power of the teacher but by the thoughtfulness of their ideas.


English: William Shatner photographed by Jerry...
English: William Shatner photographed by Jerry Avenaim (Photo credit: Wikipedia)


TekWar by William Shatner – His (first?) science fiction novel (more of detective novel) about a detective solving some mystery in a (barely) futuristic Earth


– If someone does you a favor (like getting you out of frozen animation jail), then they probably want something from you, and thus either beware of gifts (including kind words) from students and colleagues especially when you are struggling, or you should develop those dependent relationships so that if you get in trouble, you’ll have some people to help you out.


– Success comes more from who you know, even if you know them from not so great experiences (and maybe those are more valuable that those from positive experiences). So those difficult moments in teaching probably contribute more to student learning than all the highly developed super-smooth moments combined.


– If you are famous and write a not-so-good book make it a quick read. If you are teaching something that maybe you aren’t so good at teaching, tell the story quickly.  If you drag it out, you are just going to remind people that it isn’t so good; in looking back at the book (which I did actually complete), I might actually recommend it to someone (not a close friend, but someone).




creativity (Photo credit: Sean MacEntee)


The Creative Experience, ed. by Stanley Rosner and Lawrence Abt – A collection of conversations with 23 distinguished (in 1970) scientists and artists about creativity in their fields.


– Creativity can come from deliberate actions but more likely comes out of spontaneous events and semi-negative situations (e.g. boredom or frustration). If we think of learning as ‘making sense’ then it has strong ties to creativity and thus well-organized/smooth events might not be the best for learning. It may be more productive to allow or even create some spur of the moment and/or frustrating experiences.


– Those who are viewed as being creative are often less aware of what creativity looks like. In the book, the interviews with the scientists were more thoughtful and informative about creativity than those from the artists.  The artists, being in a creative field, can separate what they do in general, from what they do in particular that involves creativity. So it may also be that a natural teacher, or one who has received good evaluations without much effort might not have really much to say about being a good teacher as they might not be able to tell what they do is really about being effective and not just something that they do.  A teacher that has struggled to become better (and has done so) probably has much more to say about effective teaching.


So if you don’t check out your local thrift stores for books, you should. Also, read a variety of books as they can contribute some good ideas about teaching.


Testing Procedure

An opinion on how one should or should not do testing in a math class.

[Updated to add justifications]

There’s no particular reason for this topic as I’m not teaching at the moment and I’ve not had to deal with anyone else’s testing issues. It just came to mind and I thought I’d get the ideas out.

Here are my shoulds for a major math test (think midterm or final, not quiz):

  1. It should be out of 100 points. Because it’s easy to understand what the grade means and you have enough room for decent ranges for all the grade groups.
  2. It should be designed so that it can be graded in whole points; no fractions or decimals. Because if you open the door for fractions, then you just encourage arguments about grades.
  3. It should be designed so that the average grade is in the 70-80 range (depending on the course and class, of course), and with a reasonable distribution. Because this eases interpretation, avoids crazy curves and gives enough room for distinctive (by grades) scores.
  4. It should be comparable in style and depth to the way the course has been presented so far. Because that’s the only fair thing.
  5. It should contain a range of problems covering the topics of the test.  Because that’s sort of obvious, but also because the score will be interpreted as the knowledge of all the material.
  6. It should be take about, but not more than, the allotted time for most students to complete, with an opportunity for quickly checking answers. Because if time becomes too big of a factor, then you are testing other skills and you’ll get strange results (and lots of complaints).

And here are some should nots:

  1. It should not have unnecessary multi-part questions, i.e. where the answer to part a. is needed for part b. Because then you are stuck deciding what to do with tests where they get a. wrong, and in general it becomes much harder to connect performance to learning.
  2. It should not overemphasize any aspect of the course or prerequisite knowledge (like algebra for calculus). Because you just have to accept that they’ve passed the prerequisite and not keep punishing them for not making an A in it.
  3. It should not contain any problems that are intentionally tricky. Because … do I really have to have a reason for this?
  4. It should not be used to introduce new material to the students. Because, a test is a summary assessment and the pressures and the results of the test don’t relate well to students learning during the experience.
  5. It should not include extra credit problems to make up for excess difficultly elsewhere. Because then the distribution of the problem emphasis and difficulty is out of balance, and you are putting too much on a students ability to figure out which problems they can and can’t do.
  6. It should not be used to reward or punish any set of students, e.g. including a special problem covered only on a day that a large portion of the class missed. Because it’s obviously unfair and a bad idea and …

I can’t say that I always follow these, but at least it’s important to be aware of what I’m doing and have some reason for it. And I can’t say that not following these guarantees a bad test. However, it’s not a bad place to start when designing a test, or at least to think about what you expect before designing a test.

Learn is to Teach, as Unlearn is to ?

One of the big issues in college education is helping students unlearn things from before that either block additional learning or are just plain wrong; how can we unteach them?

An example from soccer: It is often difficult to get a young player to be able to take a successful shot on the goal. Some figure it out on their own, but most have the problem of always kicking the ball to the goalie. One of the basic causes is that early players learn to kick a soccer ball by getting with a partner and kicking it back and forth directly to each other.  Getting the ball directly to the other player is the goal of the exercise, and so the young player learns to always try to kick the ball to the other player.  You can try to help the player by developing new skills, by telling them to not look at the goalie when they take the kick.  This might work, but it doesn’t address the basic skill they learned, and that skill also makes other aspects of the game less successful (e.g. they pass directly to a player, instead of out into the space in front of the player, and they want to accept a pass by facing the other player rather than in a position to keep running forward with the ball). So some coaches have developed some different drills to help players develop better skills that lead to more productive plays (like passing while running around in a circle).

Using this example as a model, we see that once we’ve identified something that needs to be unlearned, we should avoid adding a new set of behaviors to fix it, but instead try to identify the source idea and how it was learned and try to develop some new learning exercises to replace the one bad source with a good source. What the example doesn’t resolve is how to convince the students to change and how to figure out what sort of approach will lead to change.

  • Identify the thing to be unlearned. Students will always make mistakes, and so we have to look for repeated mistakes, and then dig deeper to see what’s causing the mistakes. We have to eliminate the usual problems that happen when learning something new, and focus on cases where they repeat the mistake because, in some sense, they think they are doing the right thing. These may only show up in high stress situations like tests, but we have to separate anxiety issues from ideas that need to be unlearned.  These ideas might also show up in areas where there’s initially success, but then problems show up in extensions of the ideas. We probably can’t rely on students to self-identify these ideas, because either they will think they are correct, so won’t think that they are the problem or they will associate the mistakes they make with more common learning issues.
  • Find the source of the idea to be unlearned. This challenging especially if the student thinks that the idea is correct or has been successful with it at some level. It helps if you have multiple wrong ideas that you can trace back to the same source, but if you just have one, you’ll have to be a good detective to find the source. If it is a common error, there might be literature on it. It may be that you can’t find a source and just have to proceed on and hope that as you try to address the idea, the source will turn up.
  • Devise lessons to address the source or at least a broader version of the idea. This should be done joint with helping the student identify that the idea is wrong enough to do something about it. So the first part should be in examples that show that the idea is wrong.  This can be either ones that show it directly or ones that show that as a consequence of the idea that something bad happens. Then they should do come correct examples, and have some ideas as to how to check their own knowledge. The important part is to follow up with an assessment to check deeper and persistent understanding, and ideally never let them get away with using the old idea.