A New Class

August is here and I am about to start teaching a new class, the same Math for Life Scientists that I have taught for the last four summers. This year, I want to deliberately implement thoughtful learning. Here are my plans, pristine before their collision with reality.

• I'm going to pause more. It turns out that the "what just happened?" processing time I started using last fall based on intuition is actually a technique called the "pause procedure" with a fairly solid research base dating back to the 1970s. (See, for example, this and this.) In the original studies, students were just asked to compare their notes with a neighbor's, although some instructors assign more complex tasks. Since time is an absolute requirement for thoughtful learning, the pause procedure fits this approach perfectly. I plan to mostly use pauses to have students compare notes or discuss confusing points, as in the original studies, and explicitly ask them to come up with questions.
• I'm going to avoid multiple choice questions and the use of peer discussion as a fallback when no one answers a question. These strategies resulted in the least participatory classes I ever taught and had no discernible effect on student learning. There's (probably) nothing wrong with peer discussion, but only if it's planned from the start. Otherwise, it rewards nonparticipation. If students are truly stumped by one of my questions, I can scaffold.
• I'm going to do even more retrieval practice, especially asking students to write summaries at the end of class. If it's not too much trouble, I might try the three-step "brain, notes, other students" color-coded approach, but that might be too much to fit into an already packed summer schedule.
• Finally, I'm going to try to give students more interesting conceptual questions to think about during class and breaks. This means explicitly asking them to not read the book before it's assigned. That shouldn't be a hard sell.

 We'll see how this goes.

Training for Flexible Teaching

For about the last year, I've been working with a speech and voice coach and recently started taking an acting class she teaches at UCLA Extension. Halfway into the course, I am most struck not by any particular technique or exercise but by how many ways there are to do one thing.

Several weeks ago, we started learning short monologues. We then explored each in three different ways ("body NRGs" in the jargon of the method we are using). Sometimes, the teacher asks students performing the monologues to do them in a different way, saying, "What if your director asks for something completely different?" Many of these experiments produce results that are unexpected but make sense. Even the ones that don't often reveal something about a piece that wasn't apparent otherwise. The point is to explore many possibilities before settling on one and be able to respond creatively to whatever happens.

In math and science teaching, at least in higher education, we often seem to look for the One Best Way to teach a topic. But life interferes. Sometimes, you write out careful notes and a student asks an insightful question, which leads to a twenty-minute discussion. Sometimes, your planned ten-minute review becomes the whole lesson because that's what the students turn out to need. Sometimes, you approach a student to help them and find that they are too frustrated or upset to focus. Being a good teacher means being able to respond to these circumstances in the moment, reacting flexibly while still accomplishing what you need to accomplish. This means that teacher preparation at every level, from formal coursework to writing out your notes the night before a class, needs to focus on developing flexibility. The point of preparation is not to know exactly what you will do but to be able to respond to whatever comes up.

Since I'm not teaching this quarter, I'm trying to implement a flexibility-building type of preparation with the undergraduate learning assistants I supervise. Right now, I'm just trying to ask them for multiple possible problems and solutions that might come up as they help students. In the future, I may develop more methods, but training for flexibility looks like a good concept.

Time to Think

What do the following two scenarios have in common?
1. A professor gives a dense, fast-paced lecture with lots of slides. Students scribble down notes, trying to keep up. They need to get all the key information down before class ends.
2. In a flipped classroom, students go from one clicker question to the next. They talk about each question with a partner, and once all the answers are in and the instructor has expanded on them, go on to the next problem. No question takes more than a few minutes to get through.

These scenarios are taken from styles of teaching that are typically held up as polar opposites, yet I would argue that they are more similar than different. In particular, they fail the same way. In neither classroom is deep thought occurring. And it is not occurring for the same reason -- lack of time.

Ben Orlin has a typically charming post on barriers to deep thinking in school. However, I think he missed one. Students do not think deeply in school because there is no time for them to do so.

The primary requirement for thoughtful learning is time because the primary requirement for thought is time -- whether for private contemplation or for a conversation to proceed beyond the obvious. I have been to too many teaching workshops where participants were given a question to discuss in groups and just as the discussion was getting good, just as learning was starting to occur, we were interrupted and had to go on to the next question. Using fewer questions might have worked better.

The humanist and educator Diana Senechal wrote about similar experiences during her teacher training in her book Republic of Noise:

Just as I started to ponder a topic, I had to move into my group and start working and talking. The work seemed superficial and rushed. It seemed, moreover, that the groups reached predictable conclusions about what they read or did. The instructor would move from group to group, listening to each discussion for a few minutes. When, at the end of class, she pulled together the insights of the day, it seemed that many of the finer points had vanished.

In order for our students to have a chance to think, we must slow down. If lecturing, remove some material that students can read on their own and give them time to process. In a math class I teach, I've experimented with pausing after a long derivation and giving the students a minute or two to think through what just happened in whatever way they need, whether doodling on paper, discussing or staring off into space. I plan to try explicitly providing time for students to come up with questions to ask after covering a topic.

Another potential strategy to give students more time to think, mentioned on Susan Cain's Quiet Revolution blog, is to ask them a question at the end of a lesson that will be discussed next time. Ideally, the question should be one that deserves the time and solitude this approach provides. This is quite similar to inquiry-based learning in math and may be one of the reasons I like that approach.

Let's take the time to think of ways to give our students time to think.

Two Ideas for Promoting Transfer

Often, one of the hardest things for students to do is apply a newly learned skill in a new setting, even one that looks almost the same to the teacher. The technical term for this is transfer and promoting transfer is one of the most difficult tasks in education.

The key to transferring knowledge from one situation to another is noticing that, despite superficial differences, the two situations are somehow the same on a deeper level -- in technical terms, they share the same deep structure. For example, an arms race and the ice-albedo feedback loop that enhances warming at the poles are both situations where a change in some quantity (the amount of weapons owned by country A, the amount of ice at the north pole) leads to a further change in the same direction as the initial one. Both are positive feedback loops. Transfer would involve a student who learned about positive feedback loops in the context of arms races applying their understanding to climate change, or vice versa.

Two recent papers have described promising results in promoting transfer. One is an elaboration of methods that many teachers already use, while the other is fairly new (although we sometimes use a very similar one in LS 30 and related courses).

The first paper describes something called concreteness fading. It's exactly what it sounds like -- starting with a concrete example of a topic and then gradually moving toward a fully abstract one. In this particular study, the researchers taught second- and third-graders about equivalence problems of the type 2+5+3 = 2 + __. The teaching was done either through concrete examples (sharing stickers and making balances balance), paper-and-pencil math problems, or a concreteness fading condition that started with stickers and balances, then moved to paper representations of these things, and then moved to actual problems with numbers.

 

 

 

From http://www.learningscientists.org/blog/2018/2/1-1

After the initial learning stage, the kids were presented with problems, including word problems, more complex than anything they had been taught. This was the transfer stage. The kids who were taught entirely using concrete methods performed worst, followed by those taught abstractly. The ones taught using concreteness fading did best.

What happened? Students who only see concrete examples have a hard time generalizing. They may not see the deep structure of what they are doing. (Using a variety of examples may mitigate this but is not always practical.) Abstract learning is general but often difficult. Concreteness fading may bridge the gap between the two, making the abstract learning more effective.

The other study gave undergraduates a classic problem that was analogous to a story they had read. Most people find the analogy difficult to see unless told to look for it. However, their performance improved substantially (from a 10% success rate to a 25% one) if they were asked to come up with a problem analogous to the one they were trying to solve before actually solving it.

This is a very practical result. In some cases, it may be enough to ask students to come up with examples of a new concept, which I already do (there are a number of such problems in Modeling Life) and could do more of. For more complex problems, perhaps including programming problems, asking students to come up with a problem analogous to what they are trying to solve could make sense.  At the very least, it's worth a try.

Why I Have a Soft Spot for Inquiry-Based Learning (in Math)

Someday, I am going to get punched for saying, "That's a terribly designed experiment" or "How could they analyze their data this way?" one too many times. My long-suffering colleagues routinely listen to me rant about papers with avoidably confounded variables, uninterpretable multiple regressions (there's a paper in the works) and pseudoreplication in the education literature. If you claim to have data supporting something, I want to look at it, pick it apart and think of five alternative explanations for it. Not surprisingly, I'm a big fan of the Kirscher, Sweller and Clark paper "Why Minimal Guidance During Instruction Does Not Work" (I found this link by typing the title verbatim into Google) and definitely not a fan of discovery learning. However, there is one exception. Sort of.

This exception, as the reader already knows, is inquiry-based learning in math. Specifically, and this is important, it is inquiry-based learning (IBL) in upper division or majors-oriented lower division college math classes. What I am about to say is not meant to apply in any other context.

Inquiry-based learning in college math consists of having students learn math largely or entirely by working through sequences of problems and proofs. The version in which all proofs are done by the students, who are not allowed to use any outside references, is often called the Moore method. Less pure versions also exist and seem to be more commonly used.


The basic framework of an IBL math course has students work on proofs outside of class. Class sessions consist mainly of having students present their work and other students critiquing it as necessary. The instructor provides the problems or theorem statements and a bit of guidance during discussions but otherwise stands back.

An IBL math class embodies thoughtful learning in a way few other teaching methods in any subject do. Thinking is the entire point. Furthermore, since most of the actual work is done outside of class, students have time to think deeply and to grapple with serious problems. There is an alternation between solitary and communal thinking that takes advantage of the strengths of both -- the concentration possible alone and the error-checking and fresh viewpoints provided by others. Indeed, this is how real scientific collaborations often work.

Also, IBL fits its subject in a way that is rarely possible in the sciences. (It does bear some resemblance to seminars and writing workshops in the humanities.) While it may not be (and probably isn't) the most effective way to teach specific mathematical content because of the high cognitive load imposed by figuring out a proof and the very real possibility of proving something without understanding it, if teaching a particular way of thinking is an important goal, IBL succeeds admirably.

There is another, idiosyncratic reason why I have a soft spot for IBL. A few years ago, I was sent to a week-long workshop on the subject. Of that week, no more than 20-30 minutes were devoted to reviewing research, most of which was on active learning in general. The rest was looking at implementation and the details of what actually happens in the classroom. Rather than using bad data to try to show that a particular method of teaching was best, the workshop leaders in effect said, "Here's a way to teach we think is good and here are ways to do it". In keeping with the (apocryphal but frequently misattributed to Mark Twain) principle that "It ain't what we know that gives us trouble, it's what we think we know that just ain't so," no data can be better than bad data because it doesn't cause false confidence in the way bad data does. Perhaps grist for a future post?

Read more at: https://www.brainyquote.com/quotes/mark_twain_109

Make Learning Objectives Honest

Lately, learning outcomes/goals/objectives have become all the rage in higher ed. (As this post is written in English rather than ed-speak, I will use these terms interchangeably.) These typically have the form, "After this lesson (or unit or class), you should be able to do X".  But does learning work that way?

As a practitioner of Brazilian jiu jitsu, I get twice-weekly immersion in the experience of being a student. During a typical lesson, the instructor explains and demonstrates a new technique and then has us try it out. Very often, the initial result is complete confusion. How do I get from the starting position to the intermediate one? Where do I grab? And which limb should I be using, anyway? (This is not just a result of my disability -- many physiotypical students go through the same thing.)

Even after the initial confusion abates and I can execute the move, I don't really know the technique. I may forget it by the next session. I may be able to do it on a cooperating partner, but what about one who's resisting? How do I set up the technique and when is a good time to use it? What small details make the difference between success and failure? Learning these things takes years. At what point can I be said to know the technique?

Academic learning is just as messy and multi-layered as physical. Coming to understand anything nontrivial takes time. Learning objectives that say, "After this lesson/activity/whatever, you should be able to do X" lie to students about the nature of learning. At most, a single learning experience or even a short sequence of learning experiences can make you slightly better at a skill or deepen your understanding of a concept. They should not promise to do more.

 Education consultant and blogger David Didau writes:

All too often our learning intentions are lesson menus; here is what you should know, or be able to do by the end of today’s lesson. Students are unlikely to do more than merely mimic the understanding or expertise we want them to master.

If instead we were to share our intention for students to struggle with threshold concepts, then we could tell them that it might take them weeks to wrap their heads around such troublesome knowledge. We could remind them that in this lesson they are making progress towards a goal and that there is no expectation for them to ‘get it’ in the next hour or even the next week...

Learning does not follow a neat, linear trajectory, it’s liminal. Students not only need to spend time in that confusing, frustrating in-between space, they need to know how important it is to stay there for as long as need be. If learning intentions rush or limit this experience then they might be doing more harm than good.

If you agree with this critique but still like the idea of explicit learning goals (or are required to write them), what else could you do? One possibility is to use thought-focused rather than action-focused language. I like the following possible goals for lessons or short sequences:

1. Introduce a concept or skill
2. Deepen your understanding of a concept or practice a skill
3. Connect a concept or skill to others
4. Extend a concept or skill to new contexts

If the purpose of writing goals is to structure a course and show students what they will be learning, consider using organizing questions. There can be a few overarching questions for a course -- one class I taught used "How do systems behave?",  "How can we use math to model biological systems?" and "How can we use models to predict behavior?". Each question can have subquestions and sub-subquestions that you can introduce at the start of a lesson. The virtue of this approach is that few things pique curiosity like a question.

There's nothing wrong with explicitly stating the purpose of a lesson. But let's do it in ways that don't mislead student about learning.

Silence is an Answer

Realization: when students don't answer a question in class, they are in fact saying something. They're saying that they don't understand the material adequately or don't feel confident in their understanding (or your question was vague). If that's the case, a constructive response would be scaffolding with simpler questions, providing more explanation, or asking for a question rather than an answer. Cold-calling just papers over the cracks.

Your thoughts?