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Peer Instruction is a method developed by Eric Mazur in Harvard, designed with a student-centered approach in mind. In a nutshell, the core of the method is that when presented with a problem, students who readily overcome the obstacles necessary to understand the solution are in a much better place to explain and convince other students of what is wrong with their reasoning, dealing with specific obstructions in acquiring the new concepts.

Could this peer-instruction method be successfully adapted for widespread use in a mathematics classroom? Could it be effectively employed as a guiding principle for a course?

On this site, we have dealt with questions regarding tutoring and class participation. This question is a combined form of those, in a sense. In real time, we seek to find out who is understanding the concepts, what misconceptions exist amongst the students, and how to get them to learn from each other.

Good answers might share experiences you have had with this method, on any scale, although direct experience with using this for an entire course is preferable. Good answers also might include opinions/research about peer learning, in general, and suggestions for how to optimize student learning via these ideas.

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    $\begingroup$ I think that any answer here will be "Yes, but it depends". Indeed, I try to encourage this in all courses on at least a small scale, telling students to work together on homework problems to learn from each other. Are you asking about implementing this on a larger scale, teaching an entire course with this method? If so, I think editing the question to reflect this will lead to better answers. $\endgroup$ – Brendan W. Sullivan Mar 21 '14 at 21:58
  • $\begingroup$ You're right, I thought on a larger scale. I have to leave now, but you can edit ahead if you are able. Thanks for the input. $\endgroup$ – Mark Fantini Mar 21 '14 at 23:00
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    $\begingroup$ I rewrote the question to reflect this, and added some applicable tags. $\endgroup$ – Brendan W. Sullivan Mar 22 '14 at 20:19
  • $\begingroup$ @brendansullivan07 Thank you! $\endgroup$ – Mark Fantini Mar 22 '14 at 20:51
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    $\begingroup$ You might want to have a look at the website of [carroll college][1]. They offer FAQ, project summaries, question libraries and other resources. [1]: mathquest.carroll.edu $\endgroup$ – Anschewski Mar 23 '14 at 15:00
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Well, yeah, why not?

Mazur's innovation is to import and adapt insights on learning and teaching from k-12 and apply them to the large college lecture. This is a hugely difficult task, because what we know works best for students is made difficult by a lot of the structures of college courses. For instance, we know that when students have an opportunity to attempt to solve a problem, they'll better appreciate an explanation of the solution. But where do you find time for that in a class that meets just twice a week? How do you support students when they get stuck on a problem when you have 40 people enrolled in your class? And won't this go slowly? How will we finish all the material on the syllabus?

I would have a lot of trouble applying what I do with 4th, 9th, and 11th graders to a university math course.

But Mazur has figured out a way to make some of these things work in a way that's palatable to the university environment. So, why not? His techniques aren't based on idiosyncratic principles. Here are some general principles of learning that his approach is consistent with:

  1. Teachers need to know what their students actually think.
  2. People better understand an explanation if they feel a need for that explanation.
  3. Conversation is a great context for thinking about a problem.
  4. Informal experience with an idea is a great preparation for formal experience with an idea.

(For more about general principles of teaching and learning, see the free NRC book "How People Learn". http://www.nap.edu/openbook.php?isbn=0309070368)

Anything that's consistent with general principles of learning/teaching is going to be good for your kids. So, go for it! Let us know how it goes.

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  • $\begingroup$ Can I add that I work in a project with 27 schools in NYC where our goal is to help teachers understand principles 1 through 4 at the high school level? $\endgroup$ – David Wees Mar 26 '14 at 10:04
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I am using a form of "peer instruction" in my university calculus classes for the first time this year. I basically use "clicker questions" such as those available from carroll college. I don't use them every class, but maybe 2-3 times a week. Usually I ask a question, give them a few minutes to respond, show the bar chart of votes, and give them a few more minutes to discuss with their neighbors and vote again. The rest of the time I teach in a more traditional "interactive lecture". I do think some variety in teaching method is a good thing in its own right, since different students learn best in different ways.

I am not using physical "clickers" but instead the free online software pingo that the students can connect to through their laptops, tablets, or smartphones. This has the advantage of not costing them any money, and additionally has a bit more functionality (e.g. it accepts numerical answers in addition to multiple-choice ones). Of course, it has the disadvantage of encouraging them to have their phones out during class, and also it doesn't (yet) give me the information of who voted, so nonparticipation is a problem.

Overall, I am very happy with the experiment and I'm going to continue doing it in the future, at least in lower-division courses. I feel that with well-chosen questions, I get a very good feel for what the students are stuck on conceptually, and it forces them to address such issues and talk them out with their peers and with me. It takes some skill and experience to be able to choose and write good questions, of course, and I'm getting better at it over time. I do have an easier time finding and writing questions for some topics than others. I would say that finding/writing good questions, and dealing with the nonparticipation problem, are the main issues that I'm struggling to overcome.

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Below are a few of my personal experiences (as a student) with this, followed by some (heavily qualified) possible lessons that can be extracted from them.

One of my high school math teachers would do the day's lesson at the board, then give us the remainder of the period to start on the night's homework. People would get going on the homework right away because they knew they could get help immediately if they were getting stuck.

A line of students would form at the teacher's desk as people got into the assignment and needed help. After a while, a second line would also form at my desk. The teacher never even acknowledged that this was happening; neither encouraged it nor interfered, nor even monitored as far as I was able to tell.

As a student, I was often able to understand what another student was confused about when that student asked a question, while the teacher (maybe because of the age/generation difference, or because the teacher was so far temporally from learning it herself that she had forgotten what was confusing about it) misunderstood the student's real question and answered something else.

Some teachers would let me break in and answer the actual question; in other cases, I would have to wait, then pretend to be confused about the same thing but ask it in such a way that the teacher would understand and address the actual source of the confusion.

One other instance I remember is in 8th grade (generally, the school year during which you turn 14, for those unfamiliar with the American system) we had a section of the class that was accelerated. One day we had a substitute who didn't know the material and I taught the lesson to the non-accelerated part of the class since we had seen it a few days previous.

[NOTE: large-ish edit follows, I just realized that some experience from grad school was perhaps more relevant than anything I'd said so far]

Finally, let me relate something from graduate school. I was pursuing a Master's degree in mathematics, and by this time had become reflective enough about education that I started actively observing myself in my interaction with the learning environment. One thing that I observed was that I was never able to keep up with lecture. Another was that mathematics is taught completely backwards from how it is naturally learned (for example: when you are working on a new concept, the first thing that is taught are axioms and definitions; when you are discovering mathematics, you never even think about defining things until you've started to see some significant relationships that you feel need a name, and axioms probably come last). But to return to the point--it hit me very hard one night at a study group that the only natural way for me to learn was to have someone else start asking questions about something that I kind of have some understanding of, at which point I would start to explain it as far as I knew, and then my brain would kick into high gear as it attempted to refine the explanation--at which point I would start actually figuring out the material for real. In theory, this would work if I could convince myself to attempt to explain something to, say, an inanimate object. In practice, the only time I do this is when I see someone struggling with the concept and I start trying to help them.

A proof is in some sense an explanation, and that did work, too. But I had to drag myself to the task of writing a proof, whereas if I could see another human being needing help with it, I would work on it until they dragged me away.

I was one of many people who hit graduate school and suddenly realize that they don't know how to study, because until then nothing had really challenged them. It was a huge breakthrough for me to see that there was a way that I could grapple with the material that was a natural and pleasant experience instead of "why did I think I wanted to pursue a degree in this at all?"-level drudgery.

If there is an overarching generalization to be made about this series of anecdotes with a self-selected mathophile (and obviously there may not be), it might be that peer instruction can happen naturally, if you don't do anything to interfere with it. Another takeaway might be my school-kid observation that the teachers often didn't understand what the kids were asking, while I (another kid) did. This experience was repeated many, many times for me, and not just in math, and I saw other students doing similar things (understanding the asker's question better than the teacher did, I mean). And, the graduate school stuff I mentioned suggest another possibility--there may be kids that only learn when they are explaining. In other words, it's possible that not only is peer instruction appropriate, it's the only optimal way for certain student/subject combinations.

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    $\begingroup$ I like this answer a lot! I'd recommend starting with a very quick description of where you are coming from ("Below are a few of my experiences with this as a student. At the end I'll wrap it up") and then at the end, using asterisks around your conclusion to make it stand out. These are just very small tips to make it easier to read your posts; I hope you don't take them the wrong way. See you around the site! $\endgroup$ – Chris Cunningham Mar 26 '14 at 18:23
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    $\begingroup$ @ChrisCunningham thanks, I always appreciate constructive criticism. Edited as you suggested. $\endgroup$ – msouth Mar 26 '14 at 18:42
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There is an emergent literature around peer tutoring used for mathematical instruction in urban settings.

A nice article to begin with is:

Walker, E. N. (2007). The structure and culture of developing a mathematics tutoring collaborative in an urban high school. The High School Journal, 91(1), 57-67. Link.

Abstract: This conceptual article describes a model of a school-based, student-led initiative that uses peer tutoring to address underachievement in mathematics. The model is three pronged: a) it suggests a site-based approach to building on existing student excellence in mathematics to drive improved student mathematics achievement; b) it seeks to address the lack of teacher knowledge about urban students and their mathematics understanding; and c) it aims to deepen existing mathematics knowledge, confidence and interest among high school students. In the article, I share analyses of the interactions among tutors, tutees, advisors, and teacher; the mathematical discourse within those interactions; and the hierarchical and collaborative relationships that emerged over time.

The piece above fits into a larger discussion of building mathematics communities.

For an article in this vein, see:

Walker, E. N. (2006). Urban high school students' academic communities and their effects on mathematics success. American Educational Research Journal, 43(1), 43-73. Link.

Finally, you can find an entire book by Erica Walker entitled Building Mathematics Learning Communities.

As you can see from the description there, the book calls for mathematics teachers and students to collaborate in building these communities, and identifies three fundamental principles:

  1. Urban students want to be a part of academically challenging environments.

  2. Teachers and administrators can inadvertently create obstacles that thwart the mathematics potential of students.

  3. Educators can build on existing student networks to create collaborative and non-hierarchical communities that support mathematics achievement.

I believe that details surrounding the third principle could be of help in answering your question here.

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When I was an undergraduate at Georgia Tech, I was hired to be an organizer for "Peer Lead Undergraduate Study" (PLUS) for some calculus courses (in fact, I was one of the first two, piloting the program). The program was so successful that it has been expanded and implemented for all the calculus courses at Georgia Tech, and a few other courses as well.

The PLUS sessions were optional, and viewed as a recitation of sorts. In this way, the standard student of a Georgia Tech Calc I class might attend 3 hours of lecture, 2-3 hours of recitation, and 0-1 hour of PLUS per week.

The PLUS sessions were student oriented and student run, with the coordinator being there almost as an icebreaker and inspirateur rather than teacher (in fact, we were explicitly told not to teach). In general, students who attended the PLUS sessions far outscored those who didn't (though this is almost certainly biased as those who attended are at least superficially more motivated). Overall, it allowed professor-led-lectures to be focused on presentation of material, TA-led-recitations to be more around harder questions and second viewings of material, and the PLUS sessions to be where students develop their problem-solving techniques and skills. From my experience, I'd say it was a moderate success, and could lead to something nice.

Similar to this, I've heard of Peer led team learning, which is a different model that has also been implemented in university math classes. The link is to the center of PLTL, which has additional resources (and which has been studied - which the PLUS model hasn't, as far as I can tell).

Something that strikes me about both is the sheer time effort. The only reason it was a successful program at Georgia Tech is because it was acceptable for students to have 7 hours of class time per week for a (four credit) math course. This would not be acceptable at my current university (where they spend 3.5 hours per week in a math course). On the other hand, in my experience I find that the typical student finishing a second semester calculus course at Georgia Tech is far more proficient in both technique and knowledge of theory than the typical student finishing a second semester calculus course at my current university.

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  • $\begingroup$ This is very interesting. Thank you for sharing! $\endgroup$ – Mark Fantini Mar 27 '14 at 1:55

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