Education ∪ Math ∪ Technology

Tag: classroom practices (page 2 of 4)

“Thin slicing” and its effect on educators.

I’m reading Malcolm Gladwell’s book, Blink.  He uses a variety of arguments to show the power of information we can receive from a very small amount of information, and the subconscious ways we make decisions very quickly.  It’s a fascinating read, I highly recommend checking it out.

One point he brings up a fair bit in the first half of the book is the value in making quick decisions based on limited information and how it can both be extremely beneficial for making decisions but that it can also be problematic occasionally.  He calls this process thin-slicing.

First, the power of thin-slicing is that it allows you to make decisions quickly.  You can also trust your instincts when making decisions about directions of programs, assuming you have the expertise in that area.  His book shares some research that shows that people who are experts in their area can make a judgement about their area of expertise in 10 seconds, just as easily as in 2 hours, or 3 days.  The amount of time necessary to analyze a situation and think about a recommendation is very short, with high levels of accuracy achieved in a short period of time.

However there is a darker side to this story.  Unfortunately in areas in which we are not experts, or where there is a great deal of cultural stereotyping, our unconscious decision making can betray us.  The messages which are broadcasted into society subliminally can cause us to make unconscious decisions and snap judgments that betray what our personal sense of morality would lead us to believe.  Essentially, we not only all judge a book by its cover, but we color all of our analysis of the book by the decisions we made when we saw the cover of the book.

Most educators in North America would probably not consider themselves racist but our society sends out messages of racial stereotypes on a regular basis.  We read statistics about how 5% of African American men are incarcerated and recognize that this is much larger than the number of people from any race, and we make presumptions about African Americans.  Of course there are lots of other examples of racial stereotypes in our society, most of which are present on television.  

Now as teachers, we are likely to believe that people are deserving of equality and we almost certainly are not consciously aware of our bias.  If asked, we will say that all of our students deserve equality, and that we should treat them equally.  We may need to give some our students more attention than others because of their individual needs, but we wouldn’t openly treat them differently.

Unfortunately, we cannot avoid making snap judgments about students based on our prior experiences, and the influence of the stereotypes in our society is strong.  These snap judgments will colour all of our interactions with our students and can prevent us from treating them as fairly as we would consciously like to do.  We may downgrade papers from students whom our cultural stereotypes say are supposed to struggle with literacy, or treat unfairly students who may differ in their external packaging (their dress and mannerisms) than their peers.

There is a solution to this problem, or at least a way to make the process of educational evaluation more fair for all students involved.  One of the stories Malcolm Gladwell talks about is how the orchestras around North America and Europe have been transformed by blind auditioning.  Apparently as recently as the 1980s and 1990s, most orchestras were predominantly filled with men, and women had difficulty advancing in this area.  Orchestras recognized this (you have to read the book to find out how they recognized the problem) and began to institute policies that required the gender and race of the musicians to be hidden during the auditions.  In only a few short years, the problem of diversity in orchestras has begun to be solved.

So what can we do as educators? Any time you evaluate student work, make sure the identity of the students is unknown to you while evaluating it.  Have students turn in their work with a number which is matched to their name (randomized for each assignment) and in electronic form to avoid recognizing hand-writing.  After you have read and graded each piece of work, match the numbers to the students and record the grades or feedback.

My guess is that if you institute this policy, you will be surprised on a regular basis of the quality of work that is produced both by your supposed superstars and your weaker achievers.  You will also begin to lose some of your bias as your professional experiences begin to overcome the initial stereotyping to which you have been exposed in society.

Problem based learning in math

How does problem based learning work anyway?  According to Wikipedia, "Problem-based learning (PBL) is a student-centered instructional strategy in which students collaboratively solve problems and reflect on their experiences."  To me this means, choose problems which will reflect your curriculum and which students want to solve.

Implementation of this in mathematics can be tricky for some topics, even contrived.  If you find yourself really stretching to make a particular concept or unit fit PBL, don’t use it, use some other strategy instead.  However for almost all topics finding a real-life problem, which the students think is interesting or at least has application in their life, is relatively easy.  This is a chance for we mathematics teachers to stretch our creative muscles but it is really important that the problem chosen is either something the students have a direct interest in, or something that they can see someone in their society needing to solve.

The model I use for PBL is this; I describe a problem that exists in our world and needs solving on a regular basis, and I give the students a starting place for solving the problem, then I guide the students through the solution (giving different amounts of advice depending on the understanding of the students). At the beginning of the year the problems are quite regimented, by the end of a school year some students can solve problems mostly unguided.

My objective is to choose problems which ideally weave many different areas of mathematics (or other subjects) into the problem itself.  For example, students were given an assignment to try and decide what the best possible choice of cell phone plan is in the Metro-Vancouver area.  The solution to this involved using linear functions to model the individual cell phone plans, graphical analysis of those linear models to try and determine the best plan for any given number of minutes, algebra to determine the exact number of minutes when different plans intersect, and of course, lots and lots of research into different cell phone plans.

As the students progress through the problem, I can feed them some ideas on how to proceed.  Different groups require different amounts of guidance, and the final product the students produce can vary greatly within a class.  I often find I teach a bunch of related skills to a problem at the beginning of a class, then let the students find the connections and decide how to use the skills during their project.  Most of the time if a given skill is useful, the students find a way to incorporate the use of that skill unto their solution of the problem.

Many of the authentic learning experiences I described in an earlier post can be turned into problem based learning.  You can review these projects and then think of ways you can find problems of your own to use.

Research based teaching

I’d like to be a research based teacher.  This means, if research comes out which is compelling and reliable and which suggests that an alternate approach to what I do will work better, then I’ll experiment and try that out.  If research tells us that people learn in a certain way, then I’ll need to look at my practices and adjust them correspondingly.

Here’s an example of one change I’ve made recently because of research I read.  The research is a meta-study of the relationship between the time between a learner makes a mistake and when they receive feedback on their mistake.  If there is more than a very small amount of time between making a mistake and getting feedback on that mistake, interference between the mistake and the feedback is likely to occur and the feedback may not be what is remembered, instead the mistake may be remembered.  "Teachers who want their quizzes to help students learn should try to arrange conditions so that students receive feedback as quickly as possible after they answer quiz questions." Kulik & Kulik, 1988

I don’t assign homework anymore of a quiz or exercise nature which doesn’t provide immediate feedback.  So this entire year, I haven’t assigned a single exercise from the textbook.  I still provide a textbook in case the students want to study, or practice with their tutor or parents, but we only use it during class-time.  During class I can roam the classroom and provide feedback when students are making mistakes, and I can make sure students are checking their answers with the back of the book on a regular basis. If I want students to practice assignments at home, they get an online self-correcting quiz.  Fortunately for me, all of my students have internet access at home.

One problem with this approach is that so much good educational research is locked up in the vaults of proprietary publishers and difficult to access. I am lucky and can access much of this material through my university, but I can imagine that this could be a major stumbling block for most teachers, and of course schools can’t afford to pay for the expensive subscriptions to all of the educational journals out there.  Just having access to the database to search through the journals is difficult.

We need to, as a profession, come up with a solution to this.  Either money has to be spent ensuring that teachers have access to the best educational research out there, or teachers need to become better researchers themselves.

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.

The value of homework

So I was at a dinner party last night, and was the only high school teacher in a room of university students and academics.  It was quite an enjoyable night, and I got to reconnect with a bunch of old friends.

Of course as a teacher, eventually the subject of what I do for a living comes up.  It’s pretty clear that everyone has an opinion of good and bad teaching. I made the declaration at one point that I no longer assign typical math homework.

Actually it’s true, I haven’t assigned a problem set from the textbook this entire school year.  This is a conscious decision, not me just being forgetful and a real reason behind based on research.  I do assign other types of homework occasionally.

I read something in the summer that changed my perspective on homework.  It was about the value of feedback when learning. I don’t remember the exact title of the article, it was in my course readings.  In any case, what the authors of the study discovered is that the length of time between when you make a mistake and when you get feedback on that mistake makes a huge difference in whether or not you remember either the correct material or the incorrect material.  

If you make a mistake and get feedback within a few minutes, chances are pretty good you’ll remember the feedback rather than the mistake.  As time goes on, the probability you remember the mistake instead of the feedback increases.  If it takes more than a day or so to get feedback on your mistake, chances are pretty good you’ll make the mistake again and forget the corrective feedback you received.

The implications of this in assigning student homework is pretty easy to see.  If I assign to be done Monday night, and some of my diligent but struggling learners do the homework that night, then I see the students on Wednesday, chances are pretty good that there’s very little I can do to correct the misunderstandings of the students for that assignment.

So I’ve stopped assigning homework which doesn’t give immediate feedback.  I’ve discovered dozens of websites which offer free online quizzes which are marked immediately and display the correct answer for the students. is especially good, it gives hints on how to solve the problem as they go wrong and keeps track of how many hints each student makes.

There are other types of homework you can assign.  Anything which forces the students into a state of active engagement with their material is good.  This could mean internet research and summarization, gathering curriculum resources, creation of online tutorials, extended project based work, etc…  The analogy here is, what types of things do you do as preparation for school?  These types of tasks are also appropriate for the students to do.

Really interesting conversation on #edchat on Twitter

I participated for the first time a couple of days ago in this #edchat phenomena happening on Twitter.  Basically the idea is, everyone heads to the search page on Twitter and starts having a conversation through Twitter using the hashtag #edchat.  The resulting conversation is recorded and has many people who can listen in on the conversation, and everyone is free to jump in if they want.

The Wordle (created on below is the result of our conversation topic – motivating students.  


The reason why we used Twitter for our conversation is less obvious I suppose.  We could have had a similar conversation using forum software or one giant chat, but both of these have drawbacks.  First, a forum is an asynchronous form of communication with significant delay between comments.  Twitter is a much faster mode of communication, and the ability to refresh to see more results in the search engine makes using Twitter closer to a real conversation speed.  Twitter beats a chat application largely because you can follow the participants of the Twitter chat after the conversation, and a permanent(ish) record of the conversation continues to exist on Twitter for other people to find (and possible join!).

This conversation was fun and engaging and involved over 110 participants and over 800 comments made.  That’s pretty impressive, and definitely shows me that Twitter is a very useful tool for personal professional development and collaboration.  When was the last time you had such an organized conversation with so many people?

Summary of ETEC 533

So my ETEC 533 course has wrapped up, and it ended up being very enjoyable, although a lot of work.  We have just finished our group assignment which includes an online portion, and an essay which justifies the choices we made in creating our online resource.

In this course we started by reviewing the theory behind using technology to when teaching mathematics and science.  We came to similar conclusions as in my other courses in the MET program,which is that basically teaching the technology should not be the goal when using it to teach other subject areas, and that one has to have a good lesson and justification for using the technology in order to make it work.  We also noted that most teachers lack the training they need to effectively use the technology they are increasingly provided.

Our next unit involved looking at three different types of technology enhanced learning experiences.  We tried out the Jasper series of videos, in which real-life problems are presented using video technology, which an advanced queuing system.  We were also shown the Web-based Inquiry Science Environment (WISE) system developed at the California University at Berkeley, which provides a framework for creating lessons and interactive activities online.  The final activity of this unit allowed us to explore My World, formerly called WorldWatcher, which allows students to analyze real-life geographic data.

The last unit of the course saw us look at a variety of different learning technologies, including visualization software (like Geometer’s Sketchpad), networked communities (like Second Life and a virtual field science lab), and finally hand-held technologies (like mobile phones and data probes attached to graphing calculators).

Two common threads through-out the course were the need for advance preparation for all of these technologies, and the wider world that is made available in the classroom through the use of these technologies.  Many of these technologies are expensive, and so only the richest of schools can afford to use many different technologies in their classrooms, and so part of this course is about deciding which technology suits the situation and the specific curricula being developed.

A third thread was the ability these technologies often provide social affordances in the learning of the students, and for constructivist learning principles to be applied.  Using this learning principle does not require much tailoring of the technologies we looked at in this course.

In general this course was very useful and interesting.  It was a lot of work, and I can’t say it was made any easier by the passing away of my father mid-course, or the operation I ended up needing to have at the end of the course, OR the full back-up I had to do which deleted the original version of this essay.  Despite all of those personal problems, I still think I learned a lot from this course, and was introduced to a lot of resources, some of which I hope to use again in my own teaching.

What about using hand-held devices in education?

iPhone in Education - http://apple.comAs Dede, C., Salzman, M.C., Loften, R.B., Sprague, D. (1999) suggest, hand held devices can be powerful tools for education, when used appropriately.  Dede, et. al. (1999) indicate that the devices allow for subjects to be "immersed" in the learning, be provided with "spatial…[and]…multisensory" cues, be motivated by the use of the technology, and finally feel "[a] sense of presence in a shared virtual environment" (Loftin, 1997).

However, the use of hand-held devices has yet to see any significant impact in secondary schools, at least as far as my personal experience goes, except in a small section of the curriculum.  This is partially due to a lack of funding in education, particularly for technologies which lack a proven track record at the secondary school level, and partially because of ignorance on the part of many senior educators as to the capabilities of the hand-held devices.

Two obvious exceptions to this minimal use, which are widely used across the affluent world, are graphing calculators and digital data collectors.  In my teaching experience, the school without graphing calculators is becoming very rare.  The use of digital data collectors has become so wide-spread that there is an entire physics modeling curriculum devoted to their use out of the University of Arizona (Hestenes, D., Jackson, J., 2003).

Devices which have multiple uses, and are generally considered ‘entertainment devices’ are seeing much less use in classrooms.  In this category, I include cellphones, music devices, personal organizers, and other analogous devices.  The reasons for this are, I think, obvious.  Simply put, educators have enough problems with classroom management these days without introducing another element of difficulty, and so many schools have banned the use of these types of devices, a notable example being the New York City department of education (Clark, A.S., 2006).

The educational value of the second of these type of devices is still being investigated.  Dede, et. al. (1999) argue that when used appropriately, these types of devices can have a tremendous impact on the learning of students, but that when used inappropriately, are little more than distraction devices.

Time will tell if these devices end up having a wide-spread use in education, but my personal suspicion is that they will join the ranks of the other media devices which have been used in schools (radios, etc…) and most schools will not use these devices to their whole potential.

Clark, A.S. (2006) School Cell Phone Ban Causes Uproar. Associated Press. retrieved on April 3rd from

Dede, C., Salzman, M.C., Loften, R.B., Sprague, D. (1999). Multisensory Immersion as a Modeling Environment for Learning Complex Scientific Concepts. Computer Modeling and Simulation in Science Education

Hestenes, D., Jackson, J. (2003). A Critical Role for Physicists in K-12 Science Education Reform. Arizona State University.

Loftin, R.B. (1997). Hands Across the Atlantic. IEEE Computer Graphics & Applications 17 (2), 78-79.

20 reasons not to use a one to one laptop program in your school (and some solutions)

We have a 1 to 1 program right now at the school I’m at, and there are a lot of problems with it.  Initially I was for the program, but I am becoming more and more against it, especially with the current way our program is run.  Let me list the problems I’ve discovered so far:

  1. Classroom management while students are "taking notes with their computers" is an issue.  I think installing a gigantic mirror at the back of the classroom would be ideal.
  2. Classroom management issues while the students are supposed to be working on exercises using the CD version of their textbook, or a calculator emulator, when in fact they are searching the internet deciding what shoes they are going to buy on the weekend.
  3. MSN Messenger, Skype, Google Chat, etc… name your poison here.
  4. Transition times between activities increase as you wait for the students to reboot/boot their computer, plug in their power cord, comb their hair etc…
  5. Exceptionally slow internet at our school since every student is actively connected to the internet all the time.
  6. Our wireless hotspots only support 15 active connections.  We have as many as 26 students in a class.  You do the math.
  7. Students don’t maintain their computers properly, leading to the spreading of malware, viruses, etc… through USB sticks.
  8. Since some students have malware installed, our network takes a hit as it has to defend itself against internal intruder programs searching the local network for active ports.  Every day I have 10-12 port scans that my firewall blocks.
  9. Students don’t keep their software up to date.
  10. Students don’t even keep the right software on their computer.  Equation editor is SUPPOSED to be standard in M$ Word, but hey some students have got it uninstalled… heck some students don’t even have a word processor on their computer.
  11. Students don’t have the same software on their computers.  For example, I have seen Firefox 2, Firefox 3, Safari 2, Safari 3, Internet Explorer 6, Internet Explorer 7, Google Chrome, Opera in action, all at the same time, in the same class.
  12. Students don’t know how to do "fill in the blank" on their computer, so class time is spent trouble-shooting rather than on instruction.
  13. Laptops are stolen, about 3% of them each semester.  Combination of laisse-faire attitude by students and poor security at the school.
  14. Students forget their laptops/power cords/brains at home/in locker/in canteen
  15. Three different operating systems in use.  Yes, some students are using Linux.
  16. Of the three distinctly different operating systems in use there are 3 flavours of Windows, 2 of Linux, and 3 of MAC currently in use.  Now I’m supposed to be an expert on all 8 of these flavours and plan my lessons for minor incompatibilities between them because why?
  17. "I just need to print out X for my Y class.  Can I go do it now during your [unimportant] lesson?"
  18. Students forget passwords, even for their own computers at times.  The most common one for the students to forget is the one for the wireless or for my classroom blog.
  19. The laptops are heavy.  Textbooks are heavy.  Some of my students have back problems already at an early age from carrying too much to and from school.
  20. Most teachers lack training on how to use the 1 to 1 program effectively.  We need time to be trained in optimal pedagogical techniques involving the use of technology, provided with classroom management strategies, and shown with some proof that the technology is worth using.

There are some simple solutions to these problems.

  1. Don’t let the students buy their own computers.  Either buy all of the computers for the students or require them to buy a specific model.  They need to be using exactly the same software, hardware, etc… 

    This is less important now that more applications are on the web or cross platform.
  2. Make the school in charge of installing software on the student computers.  This works better if they are actually the school’s computers and you are renting them out to the students for the year.  This way you can ensure that no games, chat programs, peer to peer file sharing programs, http proxy tunnel clients, etc… get installed on their computers. 

    This approach is too top-heavy. Recommendation instead is to make sure that teachers are aware of these issues, and then have them focus on effective teaching; which means helping students learn about appropriate timing.
  3. Have a way for the teacher to turn off access to the internet when they need.  Could be as simple as a light switch which turns off the nearest wireless box (have one wireless box per room, configure it to a minimum radius, maximum number of active connections).

    This seems kind of crazy now. So many of the applications we use are online. 

  4. Don’t use Windows until they can prove that it is as secure as the other Unix based systems.  Go with Linux and a bunch of open source software, or go for Mac and pay through the nose, either way works.

    We’ve had many less problems with viruses here at my current school, so I think that either virus protection software has gotten better, or Windows 7 is much more secure than Windows XP.

  5. Have some common sense when planning the layout of your classrooms.  Install electricity outlets in convenient locations, either right in the tables the students are using or on the floor.  Make sure there are enough outlets to go around.  Heck, put an ethernet cable port right next to each outlet and forget about wireless all together.

    I still agree with this one. Plan ahead. I think robust wireless networks have gotten easier to set up, and so the ethernet cables are less necessary. Still, it took us almost 6 months to get our wireless network stable.

  6. Make sure students are all given training on how to most effectively use their computers.  It is the job of a school to help students learn how to use these powerful devices, but to be honest, the typical classroom teacher isn’t up to the job, and they’ll be the first to admit it.  This training should happen in an information technology course taught as a core subject.  Each student should take this course each year they are in school.

    We integrate technology at my current school without too many issues. We are focusing on teacher training on how to use the technology which seems to be making a difference.

  7. Have a specialist who’s job it is to trouble shoot the computers and make sure they are all running smoothly.  Have students see this specialist outside of class time if possible.

    I agree with having a specialist around, but wonder, if a student’s paper wasn’t working, would we let them suffer until the end of the day to get it working again? If it’s a critical tool for learning, it needs to be working.

Don’t get me wrong, I’m a strong supporter of technology in the classroom.  I think there are some very powerful, very useful ways it can be used.  However I don’t think it is being used effectively at our school, and I often wish I had the power to can the whole program and start over again, implementing some of my suggestions above. Update: At my current school, I think we are working on improving our use of technology, and for quite a lot of people, it is being used effectively. Obviously, there is always room for improvement.

Update: I wrote this post nearly 4 years ago, when I worked in a very different school, and my own pedagogical approach was different. I think that battery life of computers has improved a lot since I wrote this, mitigating some of the issues, and that I see these more as learning experiences for students and teachers. With more applications being web based (and more applications supporting a wide variety of users), standardization of device and browser is a lot less important as well. Further, students will have these same issues after they leave school, so it is somewhat better for them to have them in school, where they can get some support for later in life.