Tag: The Reflective Educator (page 40 of 43)

April 11, 2011 / 3 Comments

Bad calculations

Question: If you didn’t know the procedure for addition or multiplication, and lacked numeracy skills, could you catch the errors in the calculations shown here?

This is what happens when you teach computations instead of reasoning. Anyone who looks at 14 x 5 and gets 25 does not have basic numeracy and estimation skills developed, and quite possibly has never used real objects to do multiplication before. The error is not primarily in the calculations that they are doing but in the system that leads them to trust their calculations more than their common sense and intuition about the problem they are working on. This kind of error happens independently of the tool used. If you don’t believe me that students can make similar computational mistakes using a calculator, ask math teachers how often they see 4/8 = 2 and other similar mistakes.

A mathematics curriculum based on the ability to do computations and not solve problems is flawed in my mind. We should focus on mathematics as a tool, rather than mathematics as a goal.

March 2, 2011 / 4 Comments

Cost of switching to Open License materials

I’m curious about the cost of textbooks in British Columbia, because I wondered, after doing some brief calculations in my head, how much textbooks cost in British Columbia, and how much an open source licensing system would cost by comparison. I need some help with my math, because I can’t understand why are still using the current system.

There are currently 649, 366 students listed as registered in British Columbia, according to the BC education ministry data website. If even half of those students have textbooks of some sort, and those textbooks cost an average of $25 per year for a typical 5 year replacement cycle, and if students take an average of 6 academic subjects then the total cost of textbooks in BC is about $50 million dollars. (I’m not able to look this information up easily, as most of the school districts in BC lump the cost of textbooks into their budgets for supplies). Let’s halve that number, in case some of my estimates are incorrect, which means that the cost of textbooks alone in BC is at least $25,000,000.

Of course this cost doesn’t include transporting the books, replacing lost, stolen, or damaged books, storing the books, and paying people to manage textbooks for schools or school districts.

What would an open source system cost? Imagine a system where the textbooks are written by authors, and then licensed under a creative commons license. The authors are paid a far wage for their work, but then not granted royalties once the work is complete. Please note that this is what I envision these "textbooks" looking like.

I would imagine that each course might have an author who is responsible for keeping the work up to date and maintained. If each of these authors earned $80,000 a year for their work, and we had an author in charge of each of 6 academic subjects, for about half of the students (so half of the current grades), then the total cost of the authors salaries would be $2,880,000. We’d probably want to have an chief editor of the project, so let’s add $120,000 for someone in charge of the project, to bring the salary total to $3,000,000. Further, I’m sure we would want to have a few consultants hired from time to time, expenses so that authors could attend professional development, type-setters to ensure the content looks clean, so perhaps the total budget for the project would be $5,000,000.

The authors would likely work in collaborative teams, so that although each author would be "in charge" of a specific textbook. Content changes regularly, so you would have to keep these authors on staff full time, as they would be constantly revising and upgrading their digital textbooks. The quality of the works would actually improve over the current model since any changes that needed to be made to the textbooks could be done so immediately, and then those changes pushed out to all of the digital copies of the textbooks students hold.

According to this analysis, you would save at least $20,000,000 a year using an open source model of publishing, with a conservative estimate of how much textbooks actually cost our province. I’m sure other educational districts could do similar analysis and see savings as well. Note that this savings does not take into account savings generated by not having to transport, store, maintain, upgrade, and replace textbooks.

Some other advantages of this system is that different provinces could collaborate and share units and modules of the text. We wouldn’t have to use just print resources, as I’ve argued before. Errors and omissions could be fixed on very short time-lines. The best explanation of a particular topic could be the one used, rather than relying on a single author (or small group of authors) to provide explanations. Textbooks could be shared with whomever wanted to use them. Parents and teachers could collaborate to provide translations of the textbooks.

A serious flaw with this argument is that many students in BC do not have devices capable of displaying digital textbooks and in some cases their existing devices that would work are actually banned in schools. This is an issue that needs to be resolved, however I think that $20,000,000 would go a long way toward providing students in need with some sort of electronic reader, especially if we leverage a lot of the devices students already have. (We remember that the replacement cycle on an ereader is about once every 3 years, hence we actually have about $60,000,000.)

To those who argue that this would further standardization of content across our province, you are right it would. However, currently the resources teacher use have restrictive licenses, while at least under a creative commons license educators, students, and parents could customize their textbook to suit their needs. There will always be a need for some form of a container to hold information, and for schools that container has traditionally been either the teacher or the textbook, in this system the container would more customizable and could adapt over time as our needs change.

There must be a flaw with this argument, something I’m missing. I was sure when I did these calculations I was going to end up with a different story, so please let me know what errors, omissions, and mistakes you see in my logic.

February 13, 2011 / 2 Comments

Why textbooks should be open source

In the past few years, there has been a push for open source content, and enough resources have been created so that schools can completely do away with the traditional textbook. However, adoption of open source content has been low, and the vast majority of schools are still relying on tradtional textbooks.

Here are some reasons besides "they will cost less" that school districts (and educators) should be pushing for open source content for schools.

Reduced cost for transportation if in digital form
Schools can download the textbook and (if necessary) print it out on campus as they need it, which means they can pay for bandwidth, which most schools already do, instead of paying for shipping.
Authors paid for hours worked, not for uncertain future sales
I’m an author of a textbook. It has not seen wide adoption, partially because it is a 1st edition, and partially because the market for the textbook I co-authored is pretty much saturated with a competitors product. I spent many hours writing this textbook, and have worked out that my wage for writing the textbook works out to about $2 an hour. In an open source model, you can release the book once it is published, and just pay the author for the time they’ve worked. Given the enormous savings to school districts after the restrictive license has been removed, this actually makes fiscal sense too.
Easier to keep updated since anyone can legally make revisions
Textbooks can be immediately updated as soon as our knowledge of an area increases, or if an error is found in a textbook. Compare this to the speed it takes to update a typical textbook where the only incentive to update the textbook is to increase sales.
Can be provided easily in any format
Most textbook are provided in one or two formats, meaning that once you buy a textbook, you are locked to the mode the textbook is available in, whether that is paper, or a pdf. When the textbook has an open source license, it can be legally reformated for any device.
You can be altruistic and provide curriculum to those who really need it
There are lots of places in the world which can not afford to create their own resources for their schools. Although there are cultural implications to sending them our Westernized resources, the open source license means they can customize the content for their needs. The creation of open source content is therefore also a charitable activity.
Pick and choose what you want/remixable
If the resource doesn’t work for you, you can fix it. You can mix multiple resources, and you can customize the content to whatever your needs are. Try doing that with a traditional textbook. This gives teachers more autonomy over the materials they use.
Redistribute resources
Even if you print an open source textbook, the total cost of the textbook is maybe $10. You don’t really care as much what happens to the textbook if it only costs $10. You can take all of the people and resources that were involved in the storing, shipping, and tracking textbooks and use them more efficiently elsewhere.
Doesn’t need to be just paper
A digital "textbook" could be so much more than just paper…
Collaborate between countries
As someone who doesn’t live in the US, I certainly know how much licensing gets in the way of sharing resources across the border. So often we have to wait ages for resources available elsewhere in the world to become licensed for us in Canada. With an open source license, which much more easily be transfered from one country to another, this access barrier is dropped. Now we can collaborate across cultures and across borders much more easily than ever before. This will also allow for cross-cultural exchange. Imagine being able to download chapters about the American Revolution from the US and UK perspectives.
Less work for each district
School districts can share the work for creating content. While some of these resources will be specific to a particular part of the world, much of what we teach in different parts of the world is almost exactly the same. Each school district can therefore do a little bit less work and focus on maintaining their own regional specific content.

February 1, 2011 / 4 Comments

Mumbo Jumbo

Algebra is just mumbo jumbo to most people. Seriously.

If you asked 100 high school graduates to explain how algebra works, and why it works, I’d guess that 99% of them couldn’t, not in sufficient detail to show that they really deeply understand it. Remember that I am talking about high school graduates, so these people have almost certainly had many years of algebra and algebraic concepts taught to them. Most of these people will only be able to give you some of the rules of algebra at best, and some of them don’t even remember that much.

Algebra is an amazing tool for solving problems though! Formulate a problem as an equation, and unless the equation is too complex, there is an algebraic algorithm to solve that equation, and hence the problem you formulated.

Maybe it is such a useful tool that people don’t really need to understand how it works, maybe they can get by without a deep understanding, but still be able to follow the rules of algebra and use it to solve problems. I don’t really buy that argument though, simply because people who don’t understand something are prone to make mistakes, and not be able to check their work with a reasonable level of accuracy.

Computers are also mumbo jumbo to most people. If you asked people to explain how computers work, most of them cannot. There are actually very few people in the world who can explain from start to finish how a computer works, and there is no one that can explain every single piece of a computer. Computers are still amazing tools though, and give people the ability to solve problems that would otherwise be intractable.

I think computers are a useful tool despite our lack of understanding of how they work. Like algebra, computers are a block box in which we put our inputs and get outputs and don’t understand how the inputs are related to the outputs. Given this similarity, we should look at other reasons why using a computer might be superior to algebra.

There are some significant differences between using computers to do computation, and using algebra to do computation. The first is that using a computer, the error rate is much lower. Obviously you can still press the wrong buttons, enter the wrong information, read the information the computer gives back to you improperly, so there is error, but I’d argue that this error is much less than the standard error rate for algebra. The second benefit of using computers is that they are much faster than doing even moderately complicated algebra by hand, including entering the computation into the computer. In the case that doing it by hand is faster, then I’d say you should do the calculation by hand.

The largest difference between using a computer to do the calculation and using algebra is that algebra is a single use tool. It can only be used to turn an equation into a solution. A computer can be used for so much more.

Granted we should consider computational mathematics to be a broader tool than just plain algebra, if we want a more fair comparison with a computer, but I’d argue that all of the same problems exist with other areas of computational mathematics. As we increase the scope of computations we can learn how to use, the power of the computer becomes even more evident. It takes much less effort to learn how to compute a broader scope of problems using a computer than learning all of the individual computational methods. Witness the power of Wolfram Alpha, for example. Enter in a search phrase and all sorts of useful information comes up.

So in the consideration of using computers for solving computations, over a by hand approach, we can see postulate that the computer will produce less errors, be generally faster, and is more multipurpose than the pencil and paper model is. Furthermore, the computers can do a lot more as a tool than what you can do with algebra.

Another issue I see is that our current mathematics curriculums leave very little time to learn more important skills than computation. As Dan Meyer (@ddmeyer) points out, the formulation of a problem is more important than the actual solution. Learn how to formulate problems and understand how to verify that what you are doing makes sense, then spotting errors in computation becomes that much easier. Furthermore, I’d like to see mathematics education be much more grounded in what is relevant, than be a collection of different types of math which are taught for historical purposes or because they are the ground-work for calculus.

The question for me is, why aren’t we using computers more to do mathematics in elementary and secondary education? It can’t just be because people are scared of change, can it?

January 27, 2011 / 10 Comments

The (Nearly) Paperless Classroom

I’ve been reading about people trying to implement a paperless classroom, and it occurred to me that there are plenty of things you can do to implement this type of classroom, without using a lot of technology. You don’t need a 1 to 1 laptop program at your school to make it a (nearly) paperless classroom.

First buy some whiteboard material from your local carpentry supply store. Cut it up and make pieces about two feet (60 cm) by three feet (90 cm) in size.

Here is an example of a classroom run using these whiteboards.

Having some larger whiteboards on which to share instructions or information is useful. Instead of handing out sheets of paper to students, most of what you will share will go on these whiteboards. I used to write down instructions for a project, or practice questions, or discussion ideas up on a whiteboard before school and when the particular class came in, I would put out the appropriate whiteboard. Doing this will eliminate the worlksheets from your classroom.

Next, find 3 or 4 desk-top computers to place somewhere in the back or side of your classroom, or even better separate them around the classroom so that students can crowd around them when necessary. These are your research stations and the places where students will create permanent digital copies of their work. An organization like Free Geek can help you reduce the cost of purchasing these, or you may even be able to find corporation to make a donation. It is important that at least one of these computers is reasonably decent and has an Internet connection, but the other ones don’t have to be awesome. It is amazing how much utility you can get out of an old computer when it’s running a low memory operating system like Ubuntu.

Having a document camera, or a projector hooked up to the one of the computers in the room would be useful, but not critical. You can see from the picture above that the whiteboards are large enough that when students are sharing their work, they can just hold up the whiteboard and let everyone see it. Alternatively you can treat the sharing of work as a mini-fair where each group takes a turn looking at a few other group’s work.

It would also be a good idea to equip this classroom with at least 1 or 2 digital cameras. These can be useful to take pictures of the work the students have done on the whiteboards. You can designate one of the computers as your media storage computer and upload the pictures to this computer since you will want some record of the student’s work for later.

You will also need some notebooks for the students to record other work, particularly in writing-rich classes. In some subjects you may find that the notebooks don’t see enough use to be needed, but in others they will fill up quickly. This is where most of the paper you will use in your class will be. The notebooks will be the place where individual reflection will take place and can either be shared or not shared, depending on your preference.

Another piece of the puzzle will be a library of books on the back wall, relevant to your subject area. This way students can do “off-line” research. Yes, some of the books will be woefully out of date, but if you have a variety of books, you can help kids understand that they need to examine multiple sources, and not just accept the first thoughts on a subject they find.

Most of the work students will do will happen on the whiteboards and will disappear forever after it has been erased from the boards. Some of it will be saved on the computers as a picture taken of the whiteboard. Some of it will be transcribed to the computers as you and the students decide on what summative assessments you will include.

The type of work students will do will be collaborative. Most of your assessment will be formative as you move around the room to ensure both that the students are on task, but also that they are meeting your shared expectations. Your summative assessments will either be recorded in the notebooks, or on the computers. You can use workstations as a way to differentiate your work, and to ensure that not everyone “needs” the computers at the same time.

The (nearly) paperless classroom starts with the assumption that not every piece of work students produce is worth saving forever. Most of it is just them sharing their thoughts. Think of the notebooks and workbooks your students currently have, and the notes that they take. 99% of that work will never be looked again once it is completed. It is only as small percentage of work that needs to be immortalized on paper.

Change your mindset that the paperless classroom needs a lot of technology. It doesn’t. It needs a transformation of pedagogy from teacher centred and content focused to student centred and a focus on developing skills.

Please share any other ideas you have on implementing the (nearly) paperless classroom.

January 21, 2011 / 12 Comments

How many hours do teachers work?

I just conducted a very unscientific poll. I sent out a link on Twitter only and asked people who happened to be around how many hours they worked. It’s not rigorous. However, in the limited sample group I have of 85 (update actually 132) educators on Twitter, here are the results as a CSV file. If you haven’t responded to the survey, and want to add yourself, please feel free to do so, but please be as honest as you can.

First, I had to discard some outliers. While I do believe that there are educators who work 100+ hours in a week, there are too few of you, and you are skewing my results. One lonely soul even indicated that they work 168 hours per work, or 24/7.

Number of hours educators work. Elementary: 60.5 Secondary: 58.7 Post-Secondary: 48.4 Other: 61

First, we can see that elementary school teachers work about the same as secondary school teachers. Some people responded other, and they worked 57.1 hours per week (One of the people in the "other" category contacted me to let me know he was an administrator so maybe this is true of everyone in the other category?).

Post-secondary school teachers apparently have it easy at only 47.8 hours a week, but only 8 of them responded so obviously this a pretty tiny sample size. I don’t think we can conclude much from their responses without more data.

Here’s another chart where I’ve grouped the data.

Grouped data, hours educators work

The most important message I think we can get from these graphs is that teachers work damn hard. Look at how many educators work more than 40 hours a week! Over 90% of the educators who responded to the survey indicated that they work over 40 hours a week. I still think the 3 people at 100+ hours of work a week are a bit extreme and might be exagerating slightly…

The next time someone complains to you about how long teacher vacations are, ask them to count total hours worked in a year, not weeks. You’ll come out ahead in that argument for sure.

January 13, 2011 / 26 Comments

A Fundamental Flaw in Math Education

Here’s what math curriculum looks like in most schools.

Math computation at the centre

The computations we want students to be able to do are chosen, and then we find problems that match these computations, and if we are able, we find some real life connections to the computations. Generally the real world connections are an after thought, and many times teachers are responsible for finding these connections when the textbook problems they are given are really examples of pseudo-context rather than a real connection. It can be difficult to attach real world context to the curriculum we are expected to teach and many times teachers aren’t able to do it.

There is a serious flaw in how this curriculum is constructed. The mathematics that is chosen has no motivation in the minds of the people learning it. If you have wondered why so many people hated mathematics class, it was because they couldn’t see the point of it. That’s because there was no point. People who hate mathematics were learning apparently meaningless algorithms for the sake of the algorithms themselves rather than for the processes in our world where the algorithms describe.

What if we flipped this process? What would happen if we changed the diagram so the real world problems were in the middle of the diagram, and we chose what mathematics to look at based directly on how it helped us understand the real world? The diagram would look like this:

Real world at centre

I need to be careful here to define “real world.” I do not mean that students will spend their time learning how to solve the math problems that may arise in their life like an endless stream of super-market math. What I mean by “real world” is that the problems students would work on would have a shared context to which students will understand. This context might be a real problem that students or their teachers find, it might be part of an interesting challenge given to them by a mentor, or it might be simply exploring ‘what if’ in a puzzle.

The thing is, when you place context at the centre of the curriculum an immediate shift happens. Now the mathematics itself has immediate relevance, since the applications are focused on something which has meaning for the students. You also gain the ability to shift away from a computational focus (since that’s what should be for) and look at problem formulating and solving. Gone is pseudo-context. Gone is mathematics which has no relevance in the lives of our students.

Formulating the curriculum like this also makes finding connections to other areas where the students are studying much easier. Everyone can talk about the real world, and it can happen in every subject area. It becomes easier to turn our curriculum from caged subject areas into an open dialogue about life.

It also becomes easier to update the curriculum. The topics of a typical mathematics curriculum have changed very little in the past 100 years, and are nearly universal across the globe. However, in this curriculum the focus is on what is happening in real life, and it becomes easier to select what is important to teach as all of us experience the real world daily. In fact the curriculum itself could easily change from year to year and the actual mathematics that is taught could be different in each school. After all, what is important in mathematics is the process that students go through, not the end result.

There is also room in this model of curriculum for fairly advanced topics. You can see that calculus could be one of the branches of this type of curriculum since it deals intimately with understanding many complex phenomena in our world. If a student wants to extend themselves and get excited by the mathematics itself, we can still give them that opportunity. We can be more flexible in how we plan our lessons and give students more choice in how they approach problem solving.

What is not easy to represent in this diagram is what the arrows, which are common to all of the different possible branches of mathematics and their representation to the real world content. In my mind, these arrows represent part of the process of translating a real world problem into mathematics. Students have to know how to formulate problems, develop criteria for establishing what pieces of information they have are useful, and determine if their solution makes sense. Since they know for certain that the solution represents a real world phenomena, it will be easier to judge a correct solution from an obvious false one.

This shift also makes it easier to talk about the big ideas of mathematics. Most of the time we spend our classroom time so focused on the minutia that we forget that there are some powerful ideas in mathematics that are useful tools for thinking for students.

Here is a presentation to explain some of what this shift looks like for me.

Mathematics education has to change. I have spent my life either feeling defensive about my love of mathematics, or commiserating with people who agree with me. People say that they hate mathematics because they do not see how it is relevant. Let’s change mathematics curriculum so that context (which does not necessarily have to be “real world” but should be meaningful) is king.

Update: In my current work as a curriculum developer, I’ve been working on an alternative to the proposal above in our curriculum work which embeds contexts where they are meaningful but does not flip the arrangement as above. The key shift in our curriculum work is viewing students as sense-makers and fore-fronting student thinking as much as possible while still making mathematics accessible to all students (through shifts in instructional practices rather than shifts in the mathematics itself).

January 12, 2011 / 12 Comments

School Bells Interfere With Learning

I hate being interrupted in the middle of a good learning session with my students. It has happened hundreds of times in my career because of an archaic device we use in schools known as a clock. The clock itself isn’t evil, but the way we use it in schools has serious ramifications on how our students learn.

First, because we partition students into neat packages called subjects, they are implicitly taught that learning is something we do in compartments. If you try and introduce a little bit of another subject in your subject, students object, saying "This isn’t English, Mr. Wees. Teach us Mathematics." (I’ve actually had students tell me that). Where in the real world is learning sectioned off like this? Mathematics use English (and other languages) when they explain their discoveries to other people. Biologists use geography to decide where to start their research. All of what we learn is interconnected, and more of these connections need to made obvious to the students. This is not easy to do in a school with nine 45 minute separate blocks.

Next, we tell students to stop working on a particular project when the time is up. We enforce time limits on learning! While I’ll grant that real life has deadlines and limits, it very rare indeed that someone has to complete a task "within the next 15 minutes because class will be over" (I’ve said this in my classroom, so many times I can’t keep track). Maybe you have to finish something by a particular day, or by the end of today, but you are in charge of how much you work on the subject, and not the clock. It is ridiculous the number of times I’ve seen students actively engaged in learning and have it wrecked because the end of class came. Worse, I’ve filled the last 10 minutes of a class with a meaningless activity just to ensure that I use every minute I’ve got.

We also assume that each subject area needs the same amount of time each week, and try to make sure that everyone gets their equal share of the carefully apportioned time for courses. In our school I teach IB Mathematical Studies, which requires at least 150 hours of in class instructional time. My school has carefully arranged for about 160 hours, just in case I lose some to field trips, student illness, snow days, and other time sinks. Oh right. Field trips, those banes of our teaching existence which make it so hard to plan. It’s not like any REAL learning happens during field trips anyway.

Clocks are part of the systems world of a school but they have come to rule our life world. We have let ourselves become subject to fixed schedules, daily routine, and the drudgery of a factory-like system. I’m not saying that we can do without the clocks, but maybe we need to find ways for our system to be more flexible, to allow the learning to extend when necessary, and even send off kids early for another opportunity to learn, when their lesson with us is done. Maybe we should even rethink how we schedule kids, and consider other instructional models. There are schools where there are no bells, no classes like what you would see in a traditional school, just kids (and adults) learning.

January 1, 2011 / 10 Comments

Why educators should blog: A helpful flowchart

Flow chart for teachers. Hard to reproduce as text. Sorry!

November 15, 2010 / 7 Comments

Conrad Wolfram: Teaching kids real math with computers

One of the parents of a child I taught last year shared this with me. Here is what I think the math class should look like. Forget teaching kids computation, especially when a computer can do it faster, cheaper, and more reliably.

This classroom more closely resembles Dan Meyer’s math classroom where students are expected to formulate problems but taken to a further degree. Let’s do away with the repetitive tasks that a computer can easily do by hand, make sure all students have those devices that they need to do these repetitive tasks, and then focus on how to use the computations in the real world.