Reflections on reflections

When we think about reflections in water, we usually think of calm lakes and trees on the shore opposite to us. Or clouds. Or at least that’s what I think of: Everything is so far away, that it seems to be reflected at an axis that is a horizontal line far away from us.

Then the other day I walked along Kiel Fjord and it hit me that I had never actually consciously observed reflection of things that are located close to my position, and especially things who are not pretty much equidistant to me, but where one end is a lot closer than another one. Consider the picture below: Do you notice something that looks kinda odd to you (while at the same time looking super familiar)?

2016-10-31-16-28-40

If you are wondering what I mean, I marked it in red in the picture below: The rope and its reflection! It’s embarrassing to say that (as someone who has been sailing A LOT since the age of 7) this was the first time I really noticed, but it struck me how the maximum of the parable of the reflected rope isn’t right below the minimum of the parable of the rope, but seems shifted to the left. Of course this is exactly how it should be if we think about the optics, but I was really shocked that I had never noticed before and never thought about it before! I bet if I had had to draw the reflection I would have done it wrong and probably not even noticed…

2016-10-31-16-28-40-copy

Here is another picture to show you what I mean. This is what it looks like:

2016-10-31-16-31-40

Below I’ve drawn in the original objects in blue, the axis of reflection in red and then the reflection in green:

2016-10-31-16-31-40-copy

So far, so good, everything looking the way it’s supposed to look. Right? Then look at the picture below:
2016-10-31-16-31-40-copy-2

Sorry if this seems obvious to you, but I’m fascinated with this right now :-)

But it leads to another interesting thought: Asking people to draw stuff in order to both check their understanding and also make them reflect on their understanding. I recently had the opportunity to observe a class of master students draw the SST of the mean state of the Pacific Ocean (which was an exercise that I had suggested in connection with a class on El Nino. I thought it would be neat to have them draw the mean state and then later the anomalies of El Nino and La Nina to activate prior knowledge) and it was surprising how difficult that was even though I’m sure they would all have claimed to know what the mean state looks like. Having to draw stuff really confronts us with how sure we are of things we just assumed we knew…

And then I’m pretty sure that once we’ve drawn something that we have constructed ourselves from what we knew (rather than just copied a drawing from the blackboard or a book, although I think that also helps a lot), we are a lot less likely to forget it again.

Anyway, this is a type of exercise I will use — and recommend — a lot more in the future!

Using twitter as a tool to let students discover that the topics of their courses are EVERYWHERE

This is a method that I have been excited about ever since learning about #birdclass in the “Evidence-based undergraduate STEM teaching” MOOC last year: Help students discover that the content of your class is not restricted to your class, but actually occurs everywhere! All the time! In their own lives!

The idea is that students take pictures or describe their observations related to course materials in short messages, which are posted somewhere so every participant of the class can see them.

One example where I would use this: Hydraulic jumps. As I said on Tuesday, hydraulic jumps are often taught in a way that students have a hard time realizing that they can actually observe them all the time. Most students have observed the phenomenon, maybe even consciously, yet are not able to put it together with the theory they hear about during their lectures. So why not, in your class on hydrodynamics, ask students to send in pictures of all the hydraulic jumps they happen to see in their everyday life? The collection that soon builds will likely look something like the image below: Lots of sinks, some shots of people hosing their decks or cars, lots of rivers. But does it matter if students send in the 15th picture of a sink? No, because they still looked at the sink, recognized that what they saw was a hydraulic jump, and took a picture. Even if all of this only takes 30 seconds, that’s probably 30 extra seconds a student thought about your content, that otherwise he or she would have only thought about doing their dishes or cleaning their deck or their car.

hydraulic_jumps
A collection of images, all showing hydraulic jumps of some kind.

And even if you do this with hydraulic jumps, and not with Taylor columns or whatever comes next in your class, once students start looking at the world through the kind of glasses that let them spot the hydraulic jumps, they are also going to look at waves on a puddle and tell you whether those are shallow water or deep water waves, and they are going to see refraction of waves around pylons. In short: They have learned to actually observe the kind of content you care about in class, but in their own world.

The “classic” method uses twitter to share pictures and observations, which apparently works very well. And of course you can either make it voluntary or compulsory to send in pictures, or give bonus points, and specify what kind and quality of text should come with the picture.

You, as the instructor, can also use the pictures in class as examples. Actually, I would recommend picking one or two occasionally and discussing for a minute or two why they are great examples and what is interesting about them. You can do this as introduction to that day’s topic or as a random anecdote to engage students. But acknowledging the students’ pictures and expanding on their thoughts is really useful to keep them engaged in the topic and make them excited to submit more and better pictures (hence to find better examples in their lives, which means to think more about your course’s topic!).

And you don’t even have to use twitter. Whatever learning management system you might be using might work, too, and there are many other platforms. I recently gave a workshop for instructors at TU Dresden and talked about how awesome it would be if they made their students take pictures of everything related to their class. They were (legitimately!) a bit reluctant at first, because you cannot actually see the topic of the course, measuring and automation technology (MAT), just the fridge or camera or whatever gadget that uses MAT. But still, going about your everyday life thinking about which of the technical instruments around you might be using MAT, and discovering that most of them do, is pretty awesome, isn’t it? And documenting those thoughts might already be a step towards thinking more about MAT. At least that is what I claimed, and it seems to have worked out pretty well.

We are about to try this for a course on ceramics (and I imagine we’ll see tons of false teeth, maybe some knees, some fuses, many sinks and coffee cups and flower pots, maybe the occasional piece of jewelry ), and I am hoping they will relate what they take pictures of to processes explained in class (like sintering, which seems to be THE process in that class ;-))

I am going to try to implement it in other courses, too. Because this is one of the most important motivators, isn’t it? The recognition that what that one person talks about in front of the class all the time is actually occurring in – and relevant to – my own life. How awesome is that? :-)

Have you tried something similar? How did it work out?

Student poster presentations: the cheap and easy way

Sometimes we really want our students to practice presenting posters, but we can’t afford printing all those nice A0-posters for everybody in our large class, or we don’t want them spending time on design but focus on content, or both. What then?

Well, instead of having them design A0 posters, just give them a template for 6 A4 (or letter, if you are in the US) pages (or 9, if you want more categories than I did in the example below), let them fill those with content, print them, and then either tape or pin them to a wall. Instant poster session!

Bild_Poster_English
A cheap and easy way to have students practice poster presentations

You could of course also hand them the sheets of paper that already contain the heading, or give them blank papers and let them write the titles themselves. As long as you are not interested in the design-part of creating a poster, this is a really cheap and easy way!

Will giving your students more structure make them need more structure?

One of the arguments against offering students practice opportunities online and providing automated feedback right then and there is that that way, they will never learn to work independently. Since I am working on e-assessment a lot and with many different courses at the moment, this is a fear that I definitely need to take seriously. I don’t believe that the danger is as big as it is sometimes made out to be, but I do believe that there is a vicious circle to be aware of.

It all starts with the instructor having the impression that students are not able to organize their learning on their own. Since the instructor wants the students to succeed, she offers them a clear structure, possibly with bonus points or other kinds of rewards, so they have a safe space with instantaneous feedback to practice skills that are required later. So far, so good.
Now the students are given this structure, and get used to working on problems that are presented in small portions and with instantaneous feedback. They start believing that it is the instructor’s job to organize their learning in such a way, and start relying on the instructor to provide both motivation and bite-sized exercises.
Which the instructor, in turn, notices and interprets as the students becoming less and less able to structure their learning.
At this point it is very easy to fall in the trap of trying to provide an even better, more detailed, structure, so that the students have a better chance of succeeding. Which would likely lead to the students relying even more heavily on the instructor for structure and motivation.
Teufelskreis
It is easy to fall into a vicious circle where the instructor feels like they need to provide more and more structure and motivation, and the students feel less and less responsible for their own learning.
So what can we do? On the one hand we want to help students learn our content, on the other hand they also need to learn to learn by themselves. Can both happen at the same time?
I would say yes, they can.
The first step is recognizing the danger of entering into this downward spiral. There is absolutely no point in hoping that the students will take the initiative and not fall into the trap of relying on us, even if we point out that the trap is there. Of course they might not fall in, but whether they do or not is beyond our influence. We can only directly influence our own actions, not the students’, so we need to make sure to break the spiral ourselves.
The second step is to make sure that we resist the urge to give more and more detailed exercises and feedback.
The third step is to create an exit plan. Are we planning weekly quizzes as homework that students get a certain number of bonus points for? Then we should make sure that over time, either the number of bonus points will decrease, the time interval will become longer, the tasks become more difficult, or a combination of all three. The idea is to reward the behaviour we want just long enough that students establish it, but not any longer than that.
And of course, last but not least, instead of giving students more structure, we can help them learn the tools they need to organize their learning. Be it training skills to organize yourself, or helping them find intrinsic motivation, or teaching them to ask the right questions so they can walk themselves through complex problems until they find an answer.
It’s a pretty thin line to walk, and especially the fourth step might really be out of an instructor’s control when there is a lot of content to go through in very little time and the instructor isn’t the one deciding how much time is going to be spent on which topic. Most TAs and even many teaching staff won’t have the freedom to include teaching units on learning learning or similar. Nevertheless, it is very important to be aware of the vicious circle, or of the potential of accidentally entering it, to be sure that our best intentions don’t end up making students depending on us and the structures we provide, but instead make them independent learners.

Bridging the gap between conventional mathematics teaching and the topics that engineering students are really interested in

I’m very excited to announce that I, together with Christian Seifert, have been awarded a Tandem Fellowship by the Stifterverband für die Deutsche Wissenschaft. Christian, among other things, teaches undergraduate mathematics for engineers, and together we have developed a concept to improve instruction, which we now get support to implement.

The problem that we are addressing is that mathematics is taught to 1300 students from 12 different engineering study programs at once. At the moment, in addition to lectures and practice sessions in both very large and small groups, students get weekly online exercises that they can earn bonus points with. Student feedback is positive – they appreciate the opportunity to practice, they like that they are nudged towards continuously working on whatever is currently going on in class, and obviously they like to earn bonus points they can use on the exam.
However, mathematics is not typically a subject that non-mathematicians are very keen on. Many feel like there is no relevance of the content to their lives or even their studies. And many don’t feel confident they have a chance to succeed.
As I wrote in my recent posts on motivation, both believing that you can succeed and seeing the relevance of things you are supposed to be studying to your life are necessary for people to feel intrinsically motivated. So this is where we want to start.
Since the experience with the weekly online tests is so positive, we want to develop exercises that apply the mathematics they are currently learning to topics from their own, chosen fields. So if they are supposed to practice solving a set of linear equations, students of mechanical engineering, for example, might as well use one from a mechanical engineering case. Or even better: they might be asked to develop this set of equations first, and then solve it. By connecting mathematics with topics students are really interested in, we hope to get them to engage more with matematics.
More engagement will then likely mean that they improve their understanding both of mathmatics itself and – equally important – of their main subjects, where currently manystudents lack the math skills required. At the same time, we hope this will increase student motivation for both subjects.
Of course, there is still a lot of work to be done to first implement this concept and then evaluate whether it is working as well as we thought it would, and then probably modifying it and evaluating some more. But I am excited to get started!

On grading strategies.

How do you deal with grading to make it less painful?

Talking to a friend who had to grade a lot of exams recently I mentioned a post I had written on how to make grading less painful, only to realize later that I wrote that post, but never actually posted it! So here we go now:

Last semester student numbers in the course I taught went back to less than 1/3rd of the previous year’s numbers. And yet – grading was a huge pain. So I’ve been thinking about strategies that make grading bearable.

The main thing that helps me is to make very explicit rubrics when I design the exam, long before I start grading. I think about what is the minimum requirement for each answer, and what is the level that I would expect for a B. How important are the different answers relative to each other (and hence how many points should they contribute to the final score).

But then when it comes to grading, this is what I do.

I lock myself in to avoid colleagues coming to talk to me and distract me (if at all possible – this year it was not).

I disconnect from the internet to avoid distraction.

I make sure I have enough water to drink very close by.

I go through all the same questions in all the exams before moving on to the next question and looking at that one on all the exams. This helps to make sure grading stays consistent between students.

I also look at a couple of exams before I write down the first grades, it usually takes an adjustment period.

I remind myself of how far the students have come during the course. Sometimes I look back at very early assignments if I need a reminder of where they started from.

I move around. Seriously, grading standing (or at least getting up repeatedly and walking and stretching) really helps.

I look back at early papers I wrote as a student. That really helps putting things into perspective.

I keep mental lists of the most ridiculous answers for my own entertainment (but would obviously not share them, no matter how tempting that might be).

And most importantly: I just do it. Procrastination is really not your friend when it comes to grading…

What do you think? And ideas? Comments? Suggestions? Please share!

Designing exercises with the right amount of guidance as well as the right level of difficulty

An example of one topic at different levels of difficulty.

Designing exercises at just the right level of difficulty is a pretty difficult task. On the one hand, we would like students to do a lot of thinking themselves, and sometimes even choose the methods they use to solve the questions. On the other hand, we often want them to choose the right methods, and we want to give them enough guidance to be able to actually come to a good answer in the end.

For a project I am currently involved in, I recently drew up a sketch of how a specific task could be solved at different levels of difficulty.

The topic this exercise is on “spotting the key variables using Shainin’s variables search design”, and my sketch is based on Antony’s (1999) paper. In a nutshell, the idea is that paper helicopters (maple-seed style, see image below) have many variables that influence their flight time (for example wing length, body width, number of paper clips on them, …) and a specific method (“Shainin’s variables search design”) is used to determine which variables are the most important ones.

photo-3
Paper helicopter

In the image below, you’ll find the original steps from the Antony (1999) paper in the left column. In the second column, these steps are recreated in a very closely-guided exercise. In the third column, the teaching scenario becomes less strict, (and even less strict if you omit the part in the brackets), and in the right column the whole task is designed as a problem-based scenario.

Screen Shot 2015-01-09 at 15.45.23
Flight time of a helicopter investigated at different levels of difficulty

Clearly, difficulty increases from left to right. Typically, though, motivation of students tasked with similar exercises also increases from left to right.

So which of these scenarios should we choose, and why?

Of course, there is not one clear answer. It depends on the learning outcomes (classified, for example, by Bloom or in the SOLO framework) you have decided on for your course.

If you choose one of the options further to the left, you are providing a good structure for students to work in. It is very clear what steps they are to take in which order, and what answer is expected of them. They will know whether they are fulfilling your expectations at all times.

The further towards the right you choose your approach, the more is expected from the students. Now they will need to decide themselves which methods to use, what steps to take, whether what they have done is enough to answer the question conclusively. Having the freedom to choose things is motivating for students, however only as long as the task is still solvable. You might need to provide more guidance occasionally or point out different ways they could take to come to the next step.

The reason I am writing this post is that I often see a disconnect between the standards instructors claim to have and the kind of exercises they let their students do*. If one of your learning outcomes is that students be able to select appropriate methods to solve a problem, then choosing the leftmost option is not giving your students the chance to develop that skill, because you are making all the choices for them. You could, of course, still include questions at each junction, firstly pointing out that there IS a junction (which might not be obvious to students who might be following the instructions cook-book style), and secondly asking for alternative choices to the one you made when designing the exercise, or for arguments for/against that choice. But what I see  is that instructors have students do exercises similarly to the one in the left column, probably even have them write exams in that style, yet expect them to be able to write master’s theses where they are to choose methods themselves. This post is my attempt to explain why that probably won’t work.

* if you recognize the picture above because we recently talked about it during a consultation, and are now wondering whether I’m talking about you – no, I’m not! :-)

Taxonomy of multiple choice questions

Examples of different kinds of multiple choice questions that you could use.

Multiple choice questions are a tool that is used a lot with clickers or even on exams, but they are especially on my mind these days because I’ve been exposed to them on the student side for the first time in a very long time. I’m taking the “Introduction to evidence-based STEM teaching” course on coursera, and taking the tests there, I noticed how I fall into the typical student behavior: working backwards from the given answers, rather than actually thinking about how I would answer the question first, and then looking at the possible answers. And it is amazing how high you can score just by looking at which answer contains certain key words, or whether the grammatical structure of the answers matches the question… Scary!

So now I’m thinking again about how to ask good multiple choice questions. This post is heavily inspired by a book chapter that I read a while ago in preparation for a teaching innovation: “Teaching with Classroom Response Systems – creating active learning environments” by Derek Bruff (2009). While you should really go and read the book, I will talk you through his “taxonomy of clicker questions” (chapter 3 of said book), using my own random oceanography examples.

I’m focusing here on content questions in contrast to process questions (which would deal with the learning process itself, i.e. who the students are, how they feel about things, how well they think they understand, …).

Content questions can be asked at different levels of difficulty, and also for different purposes.

Recall of facts

In the most basic case, content questions are about recall of facts on a basic level.

Which ocean has the largest surface area?

 

A: the Indian Ocean

B: the Pacific Ocean

C: the Atlantic Ocean

D: the Southern Ocean

E: I don’t know*

Recall questions are more useful for assessing learning than for engaging students in discussions. But they can also be very helpful at the beginning of class periods or new topics to help students activate prior knowledge, which will then help them connect new concepts to already existing concepts, thereby supporting deep learning. They can also help an instructor understand students’ previous knowledge in order to assess what kind of foundation can be built on with future instruction.

Conceptual Understanding Questions

Answering conceptual understanding questions requires higher-level cognitive functions than purely recalling facts. Now, in addition to recalling, students need to understand concepts. Useful “wrong” answers are typically based on student misconceptions. Offering typical student misconceptions as possible answers is a way to elicit a misconception, so it can be confronted and resolved in a next step.

 At a water depth of 2 meters, which of the following statements is correct?

 

A: A wave with a wavelength of 10 m is faster than one with 20 m.

B: A wave with a wavelength of 10 m is slower than one with 20 m.

C: A wave with a wavelength of 10 m is as fast as one with 20 m.

D: I don’t know*

It is important to ask yourself whether a question actually is a conceptual understanding question or whether it could, in fact, be answered correctly purely based on good listening or reading. Is a correct answer really an indication of a good grasp of the underlying concept?

Classification questions

Classification questions assess understanding of concepts by having students decide which answer choices fall into a given category.

Which of the following are examples of freak waves?

 

A: The 2004 Indian Ocean Boxing Day tsunami.

B: A wave with a wave height of more than twice the significant wave height.

C: A wave with a wave height of more than five times the significant wave height.

D: The highest third of waves.

E: I don’t know*

Or asked in a different way, focussing on which characteristics define a category:

Which of the following is a characteristic of a freak wave?

 

A: The wavelength is 100 times greater than the water depth

B: The wave height is more than twice the significant wave height

C: Height is in the top third of wave heights

D: I don’t know*

This type of questions is useful when students will have to use given definitions, because they practice to see whether or not a classification (and hence a method or approach) is applicable to a given situation.

Explanation of concepts

In the “explanation of concepts” type of question, students have to weigh different definitions of a given phenomenon and find the one that describes it best.

Which of the following best describes the significant wave height?

 

A: The significant wave height is the mean wave height of the highest third of waves

B: The significant wave height is the mean over the height of all waves

C: The significant wave height is the mean wave height of the highest tenth of waves

E: I don’t know*

Instead of offering your own answer choices here, you could also ask students to explain a concept in their own words and then, in a next step, have them vote on which of those is the best explanation.

Concept question

These questions test the understanding of a concept without, at the same time, testing computational skills. If the same question was asked giving numbers for the weights and distances, students might calculate the correct answer without actually having understood the concepts behind it.

To feel the same pressure at the bottom, two water-filled vessels must have…

 

A: the same height

B: the same volume

C: the same surface area

D: Both the same volume and height

E: I don’t know*

Or another example:

If you wanted to create salt fingers that formed as quickly as possible and lasted for as long as possible, how would you set up the experiment?

 

A: Using temperature and salt.

B: Using temperature and sugar.

C: Using salt and sugar.

D: I don’t know.*

Ratio reasoning question

Ratio reasoning questions let you test the understanding of a concept without testing maths skills, too.

You are sitting on a seesaw with your niece, who weighs half of your weight. In order to be able to seesaw nicely, you have to sit…

 

A: approximately twice as far from the mounting as she does.

B: approximately at the same distance from the mounting as she does.

C: approximately half as far from the mounting as she does.

D: I don’t know.*

If the concept is understood, students can answer this without having been given numbers to calculate and then decide.

Another type of question that I like:

Which of the following sketches best describes the density maximum in freshwater?

MCQ_questions

If students have a firm grasp of the concept, they will be able to pick which of the graphs represents a given concept. If they are not sure what is shown on which axis, you can be pretty sure they do not understand the concept yet.

Application questions

Application questions further integrative learning, where students bring together ideas from multiple sessions or courses.

Which has the biggest effect on sea surface temperature?

 

A: Heating through radiation from the sun.

B: Evaporative cooling.

C: Mixing with other water masses.

D: Radiation to space during night time.

E: I don’t know.*

Students here have the chance to discuss the effect sizes depending on multiple factors, like for example the geographical setting, the season, or others.

Procedural questions

Here students apply a procedure to come to the correct answer.

The phase velocity of a shallow water wave is 7 m/s. How deep is the water?

 

A: 0.5 m

B: 1 m

C: 5 m

D: 10 m

E: 50 m

F: I don’t know*

Prediction question

Have students predict something to force them to commit to once choice so they are more invested in the outcome of an experiment (or even explanation) later on.

Which will melt faster, an ice cube in fresh water or in salt water?

 

A: The one in fresh water.

B: The one in salt water.

C: No difference.

D: I don’t know.*

Or:

Will the radius of a ball launched on a rotating table increase or decrease as the speed of the rotation is increased?

 

A: Increase.

B: Decrease.

C: Stay the same.

D: Depends on the speed the ball is launched with.

E. I don’t know.*

Critical thinking questions

Critical thinking questions do not necessarily have one right answer. Instead, they provide opportunities for discussion by suggesting several valid answers.

Iron fertilization of the ocean should be…

 

A: legal, because the possible benefits outweigh the possible risks

B: illegal, because we cannot possibly estimate the risks involved in manipulating a system as complex as the ecosystem

C: legal, because we are running a huge experiment by introducing anthropogenic CO2 into the atmosphere, so continuing with the experiment is only consequent

D: illegal, because nobody should have the right to manipulate the climate for the whole planet

For critical thinking questions, the discussion step (which is always recommended!) is even more important, because now it isn’t about finding a correct answer, but about developing valid reasoning and about practicing discussion skills.

Another way to focus on the reasoning is shown in this example:

As waves travel into shallower water, the wave length has to decrease

I. because the wave is slowed down by friction with the bottom.

II. because transformation between kinetic and potential energy is taking place.

III. because the period stays constant.

 

A: only I

B: only II

C: only III

D: I and II

E: II and III

F: I and III

G: I, II, and III

H: I don’t know*

Of course, in the example above you wouldn’t have to offer all possible combinations as options, but you can pick as many as you like!

One best answer question

Choose one best answer out of several possible answers that all have their merits.

Your rosette only lets you sample 8 bottles before you have to bring it up on deck. You are interested in a high resolution profile, but also want to survey a large area. You decide to

 

A: take samples repeatedly at each station to have a high vertical resolution

B: only do one cast per station in order to cover a larger geographical range

C: look at the data at each station to determine what to do on the next station

In this case, there is no one correct answer, since the sampling strategy depends on the question you are investigating. But discussing different situations and which of the strategies above might be useful for what situation is a great exercise.

And for those of you who are interested in even more multiple choice question examples, check out the post on multiple choice questions at different Bloom levels.

* while you would probably not want to offer this option in a graded assessment, in a classroom setting that is about formative assessment or feedback, remember to include this option! Giving that option avoids wild guessing and gives you a clearer feedback on whether or not students know (or think they know) the answer.

Asking students to tell us about “the muddiest point”

Getting feedback on what was least clear in a course session.

A classroom assessment technique that I like a lot is “the muddiest point”. It is very simple: At the end of a course unit, you hand out small pieces of papers and ask students to write down the single most confusing point (or the three least clear points, or whatever you chose). You then collect the notes and go through them in preparation for the next class.

This technique can also be combined with classical minute papers, for example, or with asking students to write down the take-home message they are taking away from that teaching unit. It is nice though if take-home messages actually remain with the students to literally take home, rather than being collected by the instructor.

But give it a try – sometimes it is really surprising to see what students take home from a lesson: It might not be what you thought was the main message! Often they find anecdotes much more telling than all the other important things you thought you had conveyed so beautifully. And then the muddiest points are also really helpful to make sure you focus your energy on topics that students really need help with.

Using the “jigsaw” method for practicing solving problem-sets

A method to get all students engaged in solving problem sets.

A very common problem during problem-set solving sessions is that instead of all students being actively involved in the exercise, in each group there is one student working on the problem set, while the rest of the group is watching, paying more or (more likely) less attention. And here is what you can do to change that:

The jigsaw method (in German often called the “expert” method), you split your class into small groups. For the sake of clarity let’s assume for now that there are 9 students in your class; this would give you three groups with three students each. Each of your groups now get their own problem to work on. After a certain amount of time, the groups are mixed: In each of the new groups, you will have one member of each of the old groups. In these new groups, every student tells the other two about the problem she has been working on in her previous group and hopefully explains it well enough that in the end, everybody knows how to solve all of the problems.

This is a great method for many reasons:

  • students are actively engaged when solving the problem in their first group, because they know they will have to be the expert on it later, explaining it to others who didn’t get the chance to work on this specific problem before
  • in the second set of groups, everybody has to explain something at some point
  • you, the instructor, get to cover more problem sets this way than if you were to do all of them in sequence with the whole group.

How do you make sure that everybody knows which group they belong to at any given time? A very simple way is to just prepare little cards which you hand out to the students, as shown below:

IMG_0314
Cards to help organize the practice and discussion groups

The system then works like this: Everybody first works on the problem with the number they have on their card. Group 1 working on problem 1, group 2 on problem 2, and so forth. In the second step, all the As are grouped together and explain their problems to each other, as are the Bs, the Cs, …

Jigsaw with 9 participants

And what do I do if I have more than 9 students?

This works well with 16 students, too. 25 is already a lot – 5 people in each group is probably the upper limit of what is still productive. But you can easily split larger groups into groups of nine by color-coding your cards. Then all the reds work together, and go through the system described above, as do the blues, the greens, the yellows…

Slide2
Jigsaw with 18 participants in 6 sub-groups of 9 students going through the system as described above

This is a method that needs a little practice. And switching seats to get all students  in the right groups takes time, as does working well together in groups. But it is definitely worth the initial friction once people have gotten used to it!