Tag: The Reflective Educator (page 42 of 43)

March 26, 2010 / 1 Comment

Moderating external projects

For the past three years, I’ve been an official IB Assistant Examiner. This means that each May (or November, but I usually don’t sign up for the November sessions, too busy), I get sent a whole bunch of external exams or projects, and I have to grade the assignments. The money isn’t great, it’s a huge amount of work, but I see it as really valuable.

I had just received yet another package this morning, which one of the administrative staff gave to me, so I felt obligated to explain to her about my role as an assistant examiner for the IB. Her response was "Wow, that’s cool, it must really give you some perspective into your own students’ work."

This really is true, I love being able to see what other schools do. I can’t share it directly with my peers for confidentiality reasons, but certainly I share the principles behind how student work is arranged, and what the expectations are around the world. I’ve now observed a few dozen different school’s work, which means that I have a few dozen perspectives on what it means to produce a student project. The best part is, almost all of these projects are based on the same small set of projects, so I can actually control for type of project.

I highly recommend moderating other school’s work, the perspective you gain is totally worth it, even if the money is not.

March 3, 2010 / 6 Comments

Recreating the physical structure of a classroom

A typical classroom might look something like this. The problem with this arrangement that I see is that almost no one actually works under this arrangement. Why not? It’s distracting! this is similar to the layout in a lot of teacher staff rooms, and it is my experience that very little work happens in the staff room when it is full. There are too many people around and too many things to see and do.

Try this an experiment for your staff. Have everyone bring work for an hour and sit in an arrangement like this. Have someone sit outside the room and peek in through a window and keep track of how much individual work people do, and how often it looks like people are off task. I’m willing to bet that you will see the same thing happen under these circumstances as what happens in our classrooms. The teachers are going to start to chat with each other. It’s human nature.

So where do teachers go to get work done? Well if we are lucky we go to our own office, or we wait until we are alone after school, but for the most part, we do our important work independently from each other and without distractions. We might to discuss stuff in small groups occasionally, or chat one on one, but for the most part, we work alone. Once in a while we’ll join up and have a full group discussion with the entire staff, but rare is the school that does this more than once or twice a week.

So I propose a different arrangement. Here’s a possible variation that might work. The big difference here is, students have their own workspace. They can work in their small groups with a few students working in the middle section, possibly under the guidance of the teacher. They have much fewer distractions available to them.

As well this system preserves what I think is the best structure for when someone needs to lecture, all of the students are facing in the direction of the presenter. Of course in an ideal classroom the students are often presenting to each other and there’s nothing that stops this from happening, it just makes sure that the conversation is generally between presenter and the members of the class. For when you want to have classroom discussions with the whole group, you might book a different room with a better structure (I’m thinking a gigantic U shape would be good, or a large elliptical table).

This may not be the ideal classroom structure but personally, I think that it’s time to rethink what a classroom looks like. There should be no sacred cows in our reform of education.

Update:

What we need in schools is not one learning space or another, but more options, and more flexibility on how to use them.

February 13, 2010 / 4 Comments

Why we need to change schools

So I was struck by an interesting analogy today after reading part of a post about flipping curriculum. The problem with current education, the post claims, is that we are focusing on cramming content into courses, rather than working fundamentally on critical thinking skills. I thought, Yes, I totally agree, and then it came into focus, the reason WHY I agree.

Here’s the argument that ran through my head.

First, the amount of information that is available to be learned in our world is increasing at an exponential rate (Actually it might be increasing at an exponential TO an exponential rate, but that’s another story). We are currently attempting to decide on which of this new information is most important to be taught to students, but unfortunately we can only operate at a linear rate, which is really a fancy way of saying that each of us can only do so much work.

The process is, some experienced teachers decide on what needs to be taught during curriculum reviews which take place on the order of a few years, then this information is included in the prescribed learning outcomes for our particular part of the world. Every 5 or 6 years the curriculum gets updated. The problem becomes abundantly clear if you look at the following graph.

The blue line indicates the amount of knowledge we are able to process over the years as educators building curriculum (assuming the number of educators remains roughly constant, which in industrialized countries is approximately true) and the red line indicates the growth of knowledge over time. You may notice a huge problem is looming, very soon we will have no possible way of forcing the content based curriculum we are building match what is actually known as a species.

An analogy to this that occurred to me as a response to a Twitter post by Joe Bower, a great educator living in Alberta. He said, "How do people function properly when they follow hundreds or thousands of people on Twitter? Am I missing something?" I thought of a quick response and decided that there really isn’t a quick response and decided to write this post.

The answer is of course that you can’t possibly follow all of the information, it’s too much, so you have to rely on your ability to analyze information quickly and set limits on how long you are going to try and process information. Anyone who has followed more than a few hundred people has some trick they use to filter through the information. Some people create lists to keep track of specific users, others listen to the Twitter stream for 20 minutes at a time and rely on the fact that really useful and important information will be reTweeted. Essentially all of these people are using some sort of critical analysis of their stream to make the flow of information more manageable.

This is the critical skill we need to teach our students. It will not be possible for an individual stuck in a linear mode to be able to muster the required processing to engage meaningfully with the exponential increase of information available. Therefore in the future, everyone who wants to be successful will need to have the ability to filter information, choose reliable and useful sources of information, and build networks of people to distribute the processing of information over their personal learning network. Each person acts as a node processing part of the information, and collectively we have a chance of being able to select the most valuable information from our incredibly messy information streams.

January 28, 2010 / 0 Comments

Maybe we should be aiming for computer programming instead of calculus in Math?

I read an article one time which questioned why we choose calculus to be the top of the math pyramid in school. Basically, most of the mathematics students learn once they master the basics aims toward preparing the students to take calculus at the end of K-12 school. The article I read suggested that statistics instead of calculus should be at the top because it is much more practical to real life than calculus is.

We deliberately choose calculus to be at the top because we want our society to produce more engineers and scientists. This helped produce a generation of engineers and scientists.

However, although engineers and scientists are still needed, the US Department of Labor predicts that neither engineers nor scientists will be in the fastest growing jobs in the future. They have predicted the 30 fasted growing jobs in the United States and there is something interesting about the list. 5 of the jobs involve the use of computers. Jobs number 25, 24, 23, 4, and 1 all include the significant use of computers in a highly technical fashion. In fact all 5 of these jobs require computer programming skills to some degree.

So I propose that we make computer programming skills should be at the top of the list. This way we will be preparing our students for careers in the future rather than the careers of the past.

Now we will still end up producing engineers and scientists because there is a huge overlap between the mathematics required to master calculus and the skills required to master computer programming. We will end up producing a lot people who are totally capable of programming a computer. Students who do not end up completing the stream will still end up having a very good understanding of how a computer works, which is obviously going to be an advantage in the future anyway.

I suspect that the current stream of math would end up diverging just after algebra. It would end up involving a lot more number theory and logical reasoning and a lot less graphing and physics based mathematics (except for the stream of students interested in game programming). I don’t know that students would find this much more interesting, but at least it would pretty easy for them to use the math they were learning and use it in direct applications involving their favorite technological devices.

Maybe kids might enjoy math more?

January 20, 2010 / 2 Comments

Idea for alternate school structure – School without a daily bell schedule

So I was just walking up the steps and had an idea. What would a school without a restrictive bell schedule look like? I was wondering about this because I remember so many times this year having students working along in a great groove on one of my projects, and then suddenly time is up and the students all have to move to another room! This is very frustrating, especially if another 10 minutes means they could finish their train of thought.

So what would it take to make this work?

First, teachers would need to have daily small group planning time built into their schedules, probably every morning. They would need to plan how the schedule was going to unfold that day and to review on a daily basis the progress of the students. Technology could be used to help keep track of where students are at so that teachers don’t have to push around gigantic piles of paper. Update: Or as John Holt suggests, we could trust students more and give at least some of them more ownership over this process.

Next, the curriculum would have to be broken down, not into subject areas (except for review possibly for external certification) but by project. Each project would have to have the traditional subjects integrated into it, with percentages (and specific skills or content areas) for how much the project counts towards each subject. Students would have individualized education plans because the teachers would have the time to construct plans for each student. Update: Again, this would be easier if at least some of the students, or all of them at some stage, had more control over what they were learning, and when.

Assessment would be standards based assessment. Partially this is because it is a bit easier to assess a bunch of students who may be at difference places in your curriculum using standards, and partially because I believe that standards based assessment works better than norm referenced assessment. Finally another argument for standards based assessment is that students should move through standards, rather than through grades.

Another thing that might be possible to remove in such a school is the barriers that we construct between students of different ages. Clubs quite often have students working together with very different skill levels and ages, and quite a large number of school clubs work quite well. So it’s not impossible for students of even very different ages to work efficiently together. As well, it might be that students of different ages are working on very similar projects or even the same project (being assessed differently because the standards might be different for each student). So what I envision is a school without grade levels and maybe with a very different layout or structure.

Perhaps this school is architecturally different as well. Students would need some private space to work (maybe in multilevel groups so that older students have the responsibility to check on the younger student’s progress and model appropriate behaviour?), many small group sized rooms, and some wide open places as well. The small work spaces could be offshoots of the general meeting areas which are in turn offshoots of a larger wide open space.

Every student should have to do some physical activity each day. Physical education is SO important for children, their bodies are built to move. It would be one of the core classes in a school like this instead of an aside that is government mandated.

What else do you think this school needs? Are there any schools which are actually like this?

Update: Of course, I’ve made some pretty broad assumptions in the original version of this piece – one of which is that every student should learn exactly the same thing. While I do believe that a liberal education (in which one learns about a wide variety of things) is important, there are many, many different ways to achieve that outcome.

January 20, 2010 / 1 Comment

Authentic learning experiences

This year I have really tried to step up the process of bringing the real world into my mathematics class. A major focus has been on using technology appropriately as a tool to help solve real life problems.

Here are some examples:

Distance formula: Finding an optimal (or near optimal) solution to the Traveling Salesman problem for a small number of cities.

Basically here the students were given the assignment of choosing 6 or 7 cities fairly near each other on a Google map and finding the x and y coordinates of each city, then using the distance formula to determine the distances between the cities. Once they had this information, they were to try and figure out a shortest path, or at least something very close to the shortest path, and then justify their solution.

Linear graphs & Piecewise functions: Compare 4 or 5 difference cell phone plans.

Students should take a few cell phone plans and compare the plans, including the cost for text messages (which may include similar graphs), the cost for extras, start up costs, etc… I found the students end up needing to create piecewise functions in order to represent a cell phone plan which has a fixed rate until the minutes are used up at which point the customer has to pay extra for each minute.

Shape and Space: Design a new school building.

Here I showed the students the new lot our school is in the process of purchasing and our project is to design a building for that spot, and calculate how much their building design will cost (within the nearest $1000). It involves finding area, volumes, perimeters, scales, perspective, etc… We are using Google Sketchup for the designs but I am now trying to work out how to import the students designs into a virtual world (like OpenSim) so we can have each student group lead walk-arounds of their building.

Polynomials: Determine how many operations multiplying a 100 digit number times a 100 digit number takes.

Students are learning about computational complexity theory by analyzing the number of steps it takes to multiply numbers together. They record each step in the operation and increase the size of the numbers of each time and re-record their results. They then compare the different number of steps in each operation and try to come up with a formula, so that they can answer the 100 digit times 100 digit question. Our object: Figure out why our TI calculators can’t do this operation. It turns out that the formula itself is a polynomial, and their substitutions to check their various formulas count as a lot of practice substituting into polynomials, which was a perfect fit for our curriculum.

Quadratic functions: Create an lower powered air cannon and use it to fire potatoes a few meters.

Here the students are attempting to use quadratic math to try and analyze their cannon, then the objective is to try and hit a target with a single shot later. The cannons should be very low powered for obvious safety reasons, capable of firing a potato (or Tennis ball) a few metres at most. There is also a slight tie-in to Social Studies where my students will be studying cannons in their unit on medieval warfare.

Bearings and Angles: Set up an orienteering course in your field or local park.

Students attempt to navigate a course through a park and pick up clues at each station, which they use to figure out a problem. Students have to be able to recognize the scale on the graph, navigate using bearings, and measure angles accurately. Also lots of fun, we did this in Regents park for a couple of years in a row.

Integration: Calculate the area (or volume in a 3d integration class) of an actual 2d or 3d model.

Basically you have the students pick an object which they then find the functions (by placing the object electronically in a coordinate system) which represent the edge of the object, then place the object in a coordinate system and calculate area of the object using integration.

Percentages: Find out how much your perfect set of "gear" (clothing) costs when it is on sale and has tax added.

Students take a catalog and calculate how much it will cost for them to buy their perfect set of clothing. They can buy as many items as they want (with their imaginary money) but have to keep track of both the individual costs and the total cost of their clothing. You can also throw some curve balls at them, like if they buy more than a certain amount, they get discount, etc…

If you have any other examples of real life math being used in a project based learning context, please let me know. I’m always interested in other ideas, especially for the more challenging areas of mathematics. I’ll add more ideas here as I remember them.

November 11, 2009 / 7 Comments

Creating a WiiMote interactive white board at my school for under $50.

So someone sent me a link to the video below and I decided to act upon it. I thought I would link my resources I find for this project here, and keep you all up to date on how it is working.

The immediate attraction for this project was the idea of being able to create an interactive white board for what looks like close to $50. Our school currently has three Smartboards at the Upper school and three Smartboards at the lower school and three times that many classrooms at each building. This means that only 1 out 3 lessons is taught using a Smartboard at best, and teachers tend not to use the Smartboards. Part of the problem here is the Smartboard technology is not immediately obvious how to use, and part of it is because of training, but access to a Smartboard is a big problem for most teachers.

First I had to do some research into the controllers themselves which are necessary, which are basically a Nintendo WiiMote and a Bluetooth receiver on your laptop. According to Amazon the WiiMote costs about $35 and the Bluetooth receiver costs about $4. You might also want a cheap tripod to mount the Wiimote onto, that should cost another $15 or so and is useful but optional (my tripod cost me 9 dollars) since in the video below you can see Johnny mounts the Wiimote directly on top of the LCD projector. You also need an LED pen, which you can apparently build for about 5 dollars or purchase online. Total cost so far is 44 dollars or if you are lucky and find a cheap tripod (like I did), 53 dollars.

As for turning the Wiimote into an interactive white board, Johnny Lee (the inventor of this process) has instructions up on his website. He has also created a community forum where you can post questions, and if it is like most online communities I know, get answers to your questions.

Johnny Lee has also presented at TED, so you know there are some very smart people who love (and have tested) his idea. What I really love is the demonstration of the VR system he has designed using pretty much the same hardware with some different software.

Here’s a good video showing a pressure sensitive pen, which will make writing much easier. The brand name of the pen casing is called Tide-to-go pressure tip pen. Yes, as in Tide the laundry detergent company.

October 22, 2009 / 1 Comment

Turning Math word problems into Math video problems

Last year I tried an experiment after being exposed to research about the Jasper project. The basic idea of this project is, turn difficult word problems into authentic video problems which include potential extensions. The experiment was this, have my students create the video word problems, and start creating a library of these problems to use with my future classes.

The experience of creating the problem has some minor mathematics in it, after all the students need to formulate a difficult problem, verify that they are able to shoot the problem on video and then show a working solution to the problem (on paper or handed in separately in digital form). These skills are quite difficult, and are higher order skills in Bloom’s taxonomy.

Here’s an example of one of these word problems on the right.

It’s important to note here that there are some very difficult mathematical concepts embedded in this video. Students will need to be able to understand rate problems, solve for the distance of the falling object using kinematics, and use trigonometry to determine the distance that needs to be traveled, and then go back to rate problems to answer the question.

The whole process from start to finish took about 2 weeks (or 8 classes). One class to brainstorm the idea, one class to decide on the script and come up with the text version of the problem, and a few classes to solve the problem and do some in-class video editing. Yes, this is a lot of time, but in terms of building student self-esteem, working on very important collaboration and planning skills, it is worth it.

There’s no way that is actually enough time to produce such a high quality (for a student group) video, so I know for sure that lots of time was spent on this video outside of class, probably many hours of time.

So this process also inspires the students and gets them excited about your material. They will work much harder when they are excited about coming to class.

The video editing process itself was fairly straight forward. Most groups shot the clips with standard digital cameras, and then recorded the audio tracks after their video was done on their computers using Audacity. One group used iMovie for their editing and production, and the other 3 groups used Windows Movie Maker which was totally sufficient for their needs. If you want a no-install option, you can look at using Pixorial.com which I’ve tested out myself and works fairly well. It only really lacks two important features, the ability to edit the audio track separate from the video, and the ability to modify the video itself (instead of just moving it around), such as slow-motion, etc…

Check out these other two videos. Maybe use them with your class and try and solve the problems. As far as I remember, all of them have solutions, although some will require students to estimate distances.

March 19, 2009 / 16 Comments

Online Geogebra training

~~Hi folks,~~

I’m planning on doing an online training session, we’ll see if I get anyone to sign up! The first 20 people to post a comment here will be registered in this free training session in Geogebra. This limit of 20 people is only because http://dimdim.com restricts the free online sessions to 20 people. I’m not, by any means, an expert in the program, but I am happy to share what I have learned in 2 years of using the program.

Post a comment to this post indicating what time works best for you. You need to fill in your email field, which is hidden from everyone except me, the owner of this blog. I’ll use your email address to send you the Dimdim meeting invitation, so it is important that you include it.
~~Go to~~ http://dimdim.com and sign up for an account. You will of course need to use the same email address as step 1. You should try it out first to check to make sure there are no problems ahead of time. I’ll of course be testing this myself. Update: We might use Mikogo instead, it seems easier to use.
~~Oh and I suppose I should mention that you really want Geogebra installed. You can get it from~~ ~~http://geogebra.org~~ ~~and happily, it is free.~~
~~I’ll also be putting up some resources on the blog, linked to this post, once I get a chance to organize them. You should make sure to come back here and download those resources.~~

~~Of course this training session is dependent on my internet connection remaining up. I guess if there are problems, we can always reschedule.~~

Update:

Hello all,

A little over three years ago I decided I wanted to experiment with online learning, and I decided I would start this experiment with a training session for Geogebra, which is still software I love to use in my teaching. A week after I made announcements everywhere about the training session, my Dad died, and I never ran the session. Three years later, I’m still receiving registration requests for the program so I know there is still interest. At one point, all of you registered for this training opportunity.

I’m still personally interested in what this would look like from my perspective, but I already know of a terrific resource for learning Geogebra for beginners so I see little point in duplicating effort. Linda ran an excellent (and free) course last year, and I recommend using it as a resource for getting started with Geogebra. You can access it here: http://moodlemeets.learnnowbc.ca/course/view.php?id=3

It’s free, but you have to create a login at http://learnnowbc.ca in order to access the course. The course is archived, but there are lots of discussions archived in the forums, and you may find many of your questions answered there.

If you are still stuck, you can take a further plunge and if you have not already done so, sign up for Twitter and then search for #mathchat in the search box on Twitter. Many of the mathematics educators that post with the #mathchat hashtag use Geogebra regularly and may be able to answer your questions. Hopefully you’ve also already found http://geogebratube.com/ which already contains 10,000 Geogebra resources that are free to use and modify as you see fit.

Thank you,

David Wees

March 18, 2009 / 0 Comments

How can Geogebra be used to help students understand and visualize mathematics problems?

In your inquiry e-folio, reflect upon knowledge representation and information visualization based on your post above and the discussion it generated with your peers. Ensure that you refer to the software you chose to explore.

In my ETEC 533 class, we are in the middle of a really cool unit, and our task of this unit was to share a digital learning tool or resource with everyone else in the class. I chose to share an open source geometry program I have used a lot, Geogebra. Unfortunately my post has yet to generate any discussion, possibly because of the large number of other geometry packages available, and the therefore limited interest in this particular one.

This handy geometry package is free, cross platform, and very easy to use. It allows for the creation of geometric objects, which have various properties (including position, color, size, etc…) and which can be either a dependent or an independent object. Independent objects can have associated dependent objects, and when you modify the independent object, the dependent geometric object is modified as well.

For example, suppose we created two points in the plane as independent objects, and then created an associated line through the two points as a dependent object. When we move the position of either of the two points the line will change to match this movement. This allows students to end up with a deeper understanding of the relationship between geometric objects.

This program is very flexible, and can be used to show simple geometric relationships (like for instance the geometric fact that the sum of the interior angles of a triangle is 180 degrees) to very complex geometric properties (the limit of the sum of rectangles which approximate the area underneath a curve is equal to the exact area under the curve). Geogebra is then therefore useful in a wide variety of different contexts and branches of mathematics.

When students are using dynamic geometry software, such as Geogebra, they invariably end up with a deeper understanding of the material (Pütz 2001). This is probably because they are given a strong visual representation of the object, that comes associated with a more tactile impression that comes with using the mouse to move and adjust the object. Obviously there is a "wow" factor involved in the use of any new program, where the students are engaged with an activity simply because it is new, but it has been my experience that the use of these geometry packages ends up leading to a long lasting understanding of geometry.

Another advantage of Geogebra is that it allows the user to export the current file into a web ready format (a java applet) which can then be uploaded to a web server. This provides the ability for students and teachers to discuss and analyze each other’s work, and allows for the creation of a social discussion about the work.

Geogebra also allows a "construction protocol navigation bar" to be added to the file, which means that users can step the geometric construction process, one piece at a time. This is a tremendous advantage of Geogebra as it allows a user observing someone else’s work to have some insight into the process they went through to create it.

Geogebra allows students to actively and through the sharing of the work online, socially construct an understanding of geometry. This program allows for simple visualizations of possibly complex geometric concepts, and helps enhance a student’s understanding of those concepts.

References

Pütz, C. (2001). Teaching Descriptive Geometry: Principles and Effective Methods Demonstrated by the Example of Monge Projection, XV Conference on Graphics, Sao Paulo Brazil, November 5-9, 2001.

Hannafin, Robert D. & Scott, Barry N. (2001). Teaching and Learning with Dynamic Geometry Programs in Student-Centered Learning Environments. Computers in the Schools, 17 (1), 121-141. Retrieved March 18, 2009, from http://www.informaworld.com/10.1300/J025v17n01_10