The Reflective Educator

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Day: December 17, 2009

Integration in the real world

I’m working on a simple project to explain to students the importance of learning integration (which is an important technique in Calculus).  The basic idea is this, take a model of something from the real world, like a car for example, and find the area represented by the model.  You can talk about the importance in figuring out how much the metal will cost to make the car, or how much paint is required to cover the car, etc…

Here’s how we are going to do it in my class.

First we use our friend Google Image search and find a model.  I found some great models from http://carblueprints.info and settled on a 1931 Ford Window Coupe as an example.  It’s not clear to me what license these images are under, but given that I am using them exclusively for educational purposes and that there is no commercial value in what I am doing, I’m probably okay to use them here.

Once I had an image, I cropped it down somewhat and resized it.  This was to make it more convenient to embed in my favourite graphing program, Geogebra.  You could very easily use any graphing program here, or even print out the image and do the rest of this project on paper.

The next step was to open up Geogebra and insert the image.  Lots of tutorials on how to do this and push the image into the background, and of course the steps will vary depending on which program you use.  Once I had the image in Geogebra, I added points around the edge of the image at the critical parts of the model.  I basically want to split the edges of the model into the different functions.

Once you have the coordinate point, which are the difficult things to find really, you can now use all sorts of techniques to find the functions.  For example, which a less advanced class, they could approximate the shape given using straight lines.  For a more advanced class, they could use lines, circles, polynomials, whatever to find the functions which represent the shape of the curves.  

One nice option here is to either use the regression tool on a graphing calculator, or available in Excel to find the functions as appropriate.

Once you have the many functions which represent the shape, the students will have to find the x bounds of the functions (which should be easy if they have recorded the coordinate points) and then integrate each function over the appropriate bounds.  One thing that could get students stuck at this stage is remembering that if they actually want to find the area ABOVE the curve, they will need to either flip their function over the x-axis, or more easily, find the absolute value of their negative answer for the integration of that function.

This activity allows for all sorts of differentiation (in terms of pedagogy, not calculus!) as well.  Students who might struggle can be encouraged to choose easier models to begin with.  You can also point out that some areas of the shape might be better done using actual area formula instead of integration or potentially this same type of activity could be used at a much lower level of mathematics using just area formula.  These models represent excellent examples of composite areas, and their realism will help students recognize the relevance of what they are learning.

I haven’t actually gone through the entire process of the integration itself.  I’m going to be using this activity with my students today and I don’t want to give it all away yet!  Wait for me to post some examples of their work when they finish.

What if education followed the open source model? What would it look like?

 I’m writing this post, inspired by Tim Gower’s Massively collaborative mathematics project.  The basic premise Gowers takes in his essay is that the power of many minds, working collaboratively, is much greater than a single mind, even in a highly technical field such as mathematics.  Gowers proposed working on a particular problem in mathematics, and according to third party reports I read, the team of mathematicians that participated solved the problem in just six weeks.

So I asked myself, in what way could we replicate this process in education?  It seems to me that much of educational theory contains large and intractable problems which have many different possible theories.  It should be a perfect place to test the collective power of educators to solve problems.

I don’t know if this is exactly the right problem to start with or not, but I had a complete idea about it, so I’m starting with it.  If someone thinks some other area is more important, please produce a comment with your own complete rebuttal and we can come to some agreement about which way to proceed.

The basic rules we should follow I think should closely follow Gower’s own rules taken verbatim from his website:

1. The aim will be to produce a proof in a top-down manner. Thus, at least to start with, comments should be short and not too technical: they would be more like feasibility studies of various ideas.

2. Comments should be as easy to understand as is humanly possible. For a truly collaborative project it is not enough to have a good idea: you have to express it in such a way that others can build on it.

3. When you do research, you are more likely to succeed if you try out lots of stupid ideas. Similarly, stupid comments are welcome here. (In the sense in which I am using “stupid”, it means something completely different from “unintelligent”. It just means not fully thought through.)

4. If you can see why somebody else’s comment is stupid, point it out in a polite way. And if someone points out that your comment is stupid, do not take offence: better to have had five stupid ideas than no ideas at all. And if somebody wrongly points out that your idea is stupid, it is even more important not to take offence: just explain gently why their dismissal of your idea is itself stupid.

5. Don’t actually use the word “stupid”, except perhaps of yourself.

6. The ideal outcome would be a solution of the problem with no single individual having to think all that hard. The hard thought would be done by a sort of super-mathematician whose brain is distributed amongst bits of the brains of lots of interlinked people. So try to resist the temptation to go away and think about something and come back with carefully polished thoughts: just give quick reactions to what you read and hope that the conversation will develop in good directions.

7. If you are convinced that you could answer a question, but it would just need a couple of weeks to go away and try a few things out, then still resist the temptation to do that. Instead, explain briefly, but as precisely as you can, why you think it is feasible to answer the question and see if the collective approach gets to the answer more quickly. (The hope is that every big idea can be broken down into a sequence of small ideas. The job of any individual collaborator is to have these small ideas until the big idea becomes obvious — and therefore just a small addition to what has gone before.) Only go off on your own if there is a general consensus that that is what you should do.

8. Similarly, suppose that somebody has an imprecise idea and you think that you can write out a fully precise version. This could be extremely valuable to the project, but don’t rush ahead and do it. First, announce in a comment what you think you can do. If the responses to your comment suggest that others would welcome a fully detailed proof of some substatement, then write a further comment with a fully motivated explanation of what it is you can prove, and give a link to a pdf file that contains the proof.

9. Actual technical work, as described in 8, will mainly be of use if it can be treated as a module. That is, one would ideally like the result to be a short statement that others can use without understanding its proof.

10. Keep the discussion focused. For instance, if the project concerns a particular approach to a particular problem (as it will do at first), and it causes you to think of a completely different approach to that problem, or of a possible way of solving a different problem, then by all means mention this, but don’t disappear down a different track.

11. However, if the different track seems to be particularly fruitful, then it would perhaps be OK to suggest it, and if there is widespread agreement that it would in fact be a good idea to abandon the original project (possibly temporarily) and pursue a new one — a kind of decision that individual mathematicians make all the time — then that is permissible.

12. Suppose the experiment actually results in something publishable. Even if only a very small number of people contribute the lion’s share of the ideas, the paper will still be submitted under a collective pseudonym with a link to the entire online discussion.

In order for this to work for an educational research paper, some of the rules need to be slightly modified.  For example, in rule number 1, we need to change the word proof to something else more appropriate.  Rule #12 in my opinion should be that everyone who participates has their name attached to the study, however this may end up offending some journals given that if 500 people participate, the list of authors alone will take a couple of pages!  However most of these rules deal with the social interactions which occur for such a project, and in this sense they are applicable.  We’ll follow these rules keeping in mind that some of them will need to be slightly rewritten, and of course you are welcome to consider other ways of formulating the rules and we can come to some agreement on the reworded rules.  

My question is, "Does assigning and collecting homework in mathematics lead to greater retention of material among students?"  It’s a question which has been answered both ways many times each and which could considered to be unsolved given the controversy surrounding it.  So I’d like to try and answer it definitively.

Here is what I would outline as the steps we would need to undertake in order to answer this question.

1.  We need to recruit other people to help us with this project.  So far it has one mathematics teacher involved (myself) and will make a very weak study!

2.  Each teacher who is involved should be working on the same unit, using the same lesson plans, and assigning the same homework for the same lessons.  We want to ideally eliminate other sources of differences in student’s performance on the final test.  To this end, we need to decide on a common unit, create a pretest and a post-test for this unit, and then generate the lesson plans necessary with the accompanying homework.  Half of us will teach the same unit with the homework, the other half without.

3.  Once a teacher is ready to report results, they should do so with their name included in the results, but no other identifying information about the students.  We may even need parents to sign waivers in certain districts, although it is my understanding that there are many types of educational research that can be done without parents permission, this may be one of them.

4.  Everyone can do the units at different times, so long as they follow the same structure.  To make this more feasible, the unit we choose should be one that is taught as widely as possible and which can be independent from other units in a mathematics course.  Although as I write this I am concerned that we may be actually answering a narrower question, "Does assigning and collecting homework in this single unit of mathematics lead to greater retention of material among students?

A fringe benefit of this exercise is that everyone who participates will have all of their lessons and homework prepared for an entire unit, which may make the work of preparing results and submitting them feel a bit less onerous!

Please let me know if you want to participate in this study, remembering that everyone who participates will benefit from this study.  Also let me know if you think a different question (or procedure) is better and give me reasons why you think this is true.  My interest in this project is mostly in the WAY in which we are going to proceed rather than what we actually accomplish.