# Mathematics En Masse

So far I’ve covered a technological and an ideological problem. This one’s logistical.

Specifically, in a discipline where one question can have many answers, it’s easy to set up a traditional forum discussion where every student’s contribution is meaningful and a springboard for further interaction. One student’s opinion does not make moot the opinion of anyone else.

With mathematics, even if the question is applied and relevant, quite often there is only one correct answer. So the traditional forum method can fail, since once one student posts the answer, there is no incentive for further discussion.

This can be solved by splitting — having only individual answers or collaboration in small groups — but this removes the very thing that makes social technology exciting: collective intelligence on a scale above what is possible with in-person interaction. Imagine, for instance, having a problem not just group-solved by your individual algebra class, but by every algebra class in a school.

There’s ways around this, like:

1. Personalized data: Students might figure out the square area of a room in their house. So every student does their calculation on something different, and then statistical methods can be applied to the data as a whole.

2. Required estimates. In a mathematics modeling problem, students try to match real-world data with an equation. While there are "best-fit" equations this can be done by hand, so that there isn’t one correct answer.

Unfortunately, neither method fully taps into the power of the group dynamic. The individual contributions don’t rebound and augment each other like an open-ended discussion. So I’ll ask: what are the best ways to set up a collaborative online assignment in mathematics?

The best solution I can think of is to get students to explain process. The class could essentially write its own textbook with a wiki; the assignment one week might be to finish the online explanation of graphing parabolas, and every student has to pitch in. I have yet to try something like this, so if anyone has, drop a line in the comments.

Jason Dyer, Guest Blogger

Jason – I have a solution for this one called the “Elaboration Method” – in short. You assign a problem and everyone solves it. But the problem doesn’t end there. Then students are required to elaborate on the solution. They might show a) how to check the answer, b) how to use a graph to verify the answer c) what happens to the solution if the problem changes slightly, d) write a word problem that results in the equation or expression, e) anything that elaborates on the problem and demonstrates that they not only understood the problem, but something else that had to do with it. I’ve used this in many traditional classes and often I will get at least 10 different elaborations for the same problem. Sure, some are repeats, but many are different and the elaborations let students explore the concepts that they are learning in a very creative way.

It would not be unreasonable to post a problem set of 3-5 problems and ask students to complete an elaboration on one of the three problems. Of course, the sooner they do their elaboration, the easier it is.

I always awarded truly insightful or original elaborations with an extra point. Thus, students are rewarded for work above and beyond what is asked for.

I haven’t tried it with a wiki, but I’ve been thinking about doing it on the message boards in my calc classes. It’s a good way to emphasize conceptual understanding, notation, and the myriad of relationships between mathematical concepts.

I have a grading rubric and examples to illustrate the technique, but I have to proof 350 pages of algebra text tonight and probably won’t be able to post it today. Hopefully that will be enough to get you started…

I suggest that students work together to develop a trickiest problem possible for each chapter/lesson. Individually, students could try to develop their own “tricky” problems (you know, like the ones near the end of the set in the book; they can be solved with the same formula, but they’ve got some kind of twist or extra step). The group could then discuss each other’s problems and work on ways to combine what makes them tricky into one crazy problem. It seems like it could scale well — classes could each come up with their own and vote, or you could have smaller groups within a single class. The problems could be brought back during reviews or tests. Students who are struggling in the class would be able to give strong contributions by pointing out the specific formulas and solution steps they struggle with.

There’s something fun about students deliberately trying to make math difficult, it sort of lets them turn things around.

I love the ideas that have been shared so far. It gives me some good food for thought.

I teach elementary and often have my students work out ways to do problems on their own rather than give them the algorithm. For example, recently they found the radius, diameter, and circumference for lots of circles. They could have put their data online for everyone to see (I did it on chart paper) and consider. They used that data to begin considering ways these three measurements related. We had several discussions about it to help them, but doing it online would have been better because it would have been consistently ongoing rather than just when I stopped them all to talk.

These are all excellent ideas. Maria, I’ll be watching for that post.

A version of this has been implemented in some local classrooms to teach students how to effectively deal with test questions on state testing.

1. You give the class a release question from the mathematics test and tell them that they need to write why they picked a given answer.

2. You have them share how they came up with their answers.

3. You go through each answer and reasons to pick them with the class.

4. You discuss the distractor answers, and how to spot them.

It’s test prep, but I like the logic part of it.

Jumping off your idea on personalized data, when studying surface area and volume, we did a large scale project where students had to draw a net, measure the dimensions, find total & lateral surface area and volume, calculate the changes when dimensions were changed by certain scale factor, and so on. The catch was that each group had to find their own objects (cylinders, prisms, cubes, etc) to measure, so everyone’s results would be different. We put these on science-fair poster boards, but it could have very easily been an online collaborative project if the technology was readily available here.

Similarly, I usually have students make small, 8.5″ by 11″ “posters” that define key terms, explain how to do problems and their solutions. I hang these on the wall to help my students, but it could easily become the foundation of a wiki or other online resource that could be posted online.

A colleague of mine sent me a link to this post via del.icio.us because we love using web 2.0 tools like forums with our middle school students but we usually have the best luck in general discussions about “team issues” or “team projects.” I have been looking for new ideas on how to implement online discussion in my middle school math classes.

This post was a great read, and the discussion provided even more ideas for me to experiment with.

I have a question. How do you handle discussion in your classrooms? Do you make them private or public?

We handle discussions through Moodle making them private password protected discussions but it has taken away the aspect of publishing or interacting with the world that we really are striving for.

I will be experimenting with public discussions and blogging this next quarter at a new blog titled Mathematics in the Middle.

mathematicsinthemiddle.edublogs.org

Come check it out in a few weeks if you have a sec. and provide some feedback for a rookie.

hi awsome post, just what i am into now. I am trying to compare and contrast the effect of constrcutionist vs instructionist software in helping kids with unlike fractions. For the instructionist i am using Plato Pathways, and then am using a variety of products for constructionist part. Mainly centered around my Ning social network site. The details are in my latest blog

http://futurecollege.org/elearning.

some great help from this post though.