Sunday, November 27, 2011

More Inspiration for Math Projects

For years I have wanted to try a project-based math class. My inspiration ebbs and flows as I encounter excellent projects and rationale for executing them. Up to this point, I have left the dream as just that--a dream. There are several reasons for this:

  1. I felt I was too inexperienced to take it on.
  2. I felt the curriculum didn't lend itself nicely to projects.
  3. I didn't have the resources and infrastructure to execute it.
  4. I hadn't heard of many who believed in it.
  5. Couldn't elegantly explain why I felt it was necessary.
Slowly, over the last little while, these barriers have begun to weaken. I began by implementing smaller project work in my classes. It taught me some of the pitfalls that a fully project run class may encounter. My province went through an entire curriculum overhaul and pioneered a set of courses targeting the workplace and skilled trades. My indoctrination to twitter and blogs has provided a healthy repertoire of projects and inspiration for many more. Included with the ideas, twitter introduced me to many people who also held my vision for the potential of mathematics projects. Administrative support and my experimentation with wikis are reducing the logistical issues. All these factors culminated in a post last month where I gave my initial vision. ( It has since been shifted and tweaked as my ideas grow.

Despite all these advancements, I have not been able to sit down and write exactly why I felt that a project-based course could benefit the students at my school. (I had bits and pieces, but no unified point) This problem took a major leap toward resolution this morning when I watched a TED Talk by Dan Meyer (@ddmeyer). I have used, and modified, some of Dan's ideas in class as a litmus for project-based math; his speech perfectly embodied why I have been working toward project-based math. Watch below:

The presentation contains many valid points, but the one that struck me as the inspiration behind my vision was Dan's use of the term "intuition". I haven't been teaching for long, but have spent that time watching students hunting for answers from information sources other then themselves. All 5 symptoms mentioned in the talk could describe my students' behaviour. Even the best lessons were met with the one question that told me something had to change:

"Is this right?"

The project-based approach will not eliminate the learning of incremental skills. It does not outlaw me from taking a group of students aside to explain a particular mathematical concept. It is not about abandoning my students so they can practice formulating questions on their own. I want my class to be set-up as a launching pad for student intuition. I want to equip my students with the tools to follow their own intuition. 

In previous keynotes, I have heard the term "low-floor". Here, Meyer uses the term "level playing field". In both cases, the terms describe a math room that allows all students to become involved. Even a room, like my own, that is filled with students who think they cannot do math. Projects are built around familiar contexts to protect the dignity of all; every student has filled a container with water. Students can then use their intuition as the road map toward the solution. As they go, they will not only learn the incremental skills, they will use them in the most authentic situation possible--a situation they create. The math becomes the vocabulary for their intuition. 

The projects provide the perfect weapon against impatient problem solving. It lessens the pressure that every student feels to "play the game". My hope is to provide a setting where students are looking for a solution to a problem, but don't mind spending time doing the math along the way.


Saturday, November 19, 2011

Polling in Math Class

This past Monday I attended a professional development focused around technological infusion into our teaching. I will be the first to admit that this topic is not often tailored toward the math teachers in the building. In the morning, virtual classrooms and movie making dominated the discussions. I didn't see the implications for my mathematics classroom, until the afternoon. A facilitator introduced me to the SMS text messaging technology of polling.

I had first heard of the idea in university when certain classes included a "clicker" in the required materials. In my case, a biology professor could ask a question and instantly get a gauge on how well topics were being understood within the lecture. Polling technology essentially does the same thing by turning every student's cell phone into a clicker. 

It took me 2 minutes to set up an account at It is completely free on both ends. There is no overhead for the organizer, and the texted responses carry no charge as long as messaging is included in the phone's plan. I made this disclaimer before starting activities with the students. Many of them laughed when I explained that some phones would not be equipped with texting. (Sign of the times)

From there, creating multiple choice and open-ended polls took literally seconds. I explained the response procedure to each of my classes, and demonstrated how the results may look. This procedure took about 10 minutes per class. After that initial day, students were instructed to bring their phones to class in the event that I needed their feedback. Results were broadcasted on my SMART board.

For a quick demonstration on how to actually run the software, see the video link below. It is so intuitive that it does not take long to pick up. 

After a week of playing with the possibilities, I deciphered five immediate benefits from the technology. I over-integrated it initially to (a) make the students familiar, and (b) develop a pool of situations with which the technology is helpful. 

Five Benefits:

1) One-Hundred Percent Student Response. 
I know that teachers have a daily struggle engaging the few outliers in each classroom. Sometimes even the most potent diagnostic tools miss students. I can not begin to describe my bewilderment when a student that I have tried for months to reach pulled out their phone to reply to the poll. The results are completely anonymous; this fact, coupled with the familiarity of the technology, creates a comfortable situation for students. There is no spotlight--no hand-raising. Every student who had a phone (~95%) responded to EVERY poll. This created the most accurate diagnostic tool I have ever used in my classroom. 

2) Gauge the Class Dynamics.
Every class has its own personality, and every class encounters the material in different ways. I have two sections of Pre-calculus 20 (Grade 11) this semester. One section lost some class time due to a pep-rally and were rushed through the topic quicker. My teacher instinct told me that students in the rushed class would need more time on the topic. I set up a quick poll asking them to respond with the method of solving quadratic equations they needed the most help with. To my surprise, the case was completely the opposite. Over half of the class told me they would like help with a previous method. It took a little scrambling on my part, but their responses changed the course of their class-time. I was not discrete with this information, and students enjoyed the new-found power. One even joked that we should poll when they wanted the tests to be. 

3) Drill and Response Type Questions.
This method mimicked the clicker technology the most. The poll didn't constitute a pedagogical revolution, but rather created a safe environment where students could respond without pressure. Also, I could create the question with common pitfalls in mind. After the results were in, I was able to explain to students (in complete anonymity) why they may have arrived at a certain response. The following poll is a great example:

This was given to a class just starting the topic of slope. I knew the most common mistake would be to count the run over rise instead of the rise over run. Students also are initially confused with the nature of positive and negative slopes. When do you rise? When do you fall? Which is negative? etc. In this case, even though only 55% of students answered correctly, I was able to address everyone's problems. As the class went on, I noticed a drastic reduction in these errors because every student held ownership of a response. I also had students create polls if they finished work quickly. Not only was this a great motivator, it provided them a chance to predict common mistakes and give explanations. 

4) Open-Ended Review Tool. also allows for open ended polls where short messages are projected on the screen. I was weary of allowing students to put anything on the board with complete anonymity, but after a few glitches it became a powerful review tool.

Students were encouraged to ask questions with the poll. I told them to watch to see if others are having trouble with the same problems. Some students physically moved to be with those like-minded students. Others offered advise with the texts themselves. It reduced the amount of times I had to explain the same procedure, because those having trouble congregated. One class actually organized a study group for the next day with the poll. Students could text in for help and move on to the next problem knowing I got their request. I had a grade 11 student ask for help for the first time in 2 months using this format. I am not going to deny that there were growing pains with this freedom, but after an initial feeling-out process, it was a powerful tool.

5) Implicit Numeracy Skills.
The first question I polled was one on 'rate of change'. I hid the graph of responses while they worked and texted in (to avoid peer pressure). When I clicked the "reveal graph" button, each response had received exactly 25% of the vote. This fact was met with an audible groan from the class and spawned a VERY teachable moment. I asked the class what was wrong. They had made the immediate connection that this meant that 75% of them were wrong no matter what. I continued to poke this bear for a while. What if we had 100% on one response? Is that better or worse? What percent would you need to feel comfortable? etc.

That experience was bitter-sweet for me. On one hand, students were confused topically, but on the other, they were interpreting the data correctly. It got the class using skills that are important for any numerate citizen. 

I never imagined the immediate effect that polling would have on my class. It goes far beyond the superficial attachment that students feel to their phones. Polling did more than engage students in the lesson, it provided a connective and collaborative tissue. I look forward to finding more ways in which the technology can better the learning environment in my room. 


Saturday, November 12, 2011

Linear Functions With a Bang

Many teachers tell me that it is their creativity that limits their ability to be adaptive in the classroom. Somehow the "reform" movement (or should I say re-movement) has pigeon-holed itself into a connotation where high-energy teachers give vague tasks to groups of interested students. Out of all this, curricular outcomes explode in no particular order. This can't be further from the truth. In my view, the biggest steps toward changing student learning is changing teacher perception.

When presented with a topic to cover, there are two dominant ends of the Math-Ed spectrum. First, you have the transmission approach which carefully selects examples that represent the questions of that type that will be encountered in homework packages and on unit exams. The teacher can predict exactly what the class will look like, and students have very little control. Second, you have the open-ended approach where students are given a leading question and formulate ideas in order to solve it. In this process, the students may end up wherever their minds (and motivation) take them. The teacher has almost no clue where the class is going, and the students are given ultimate control. If a teacher is presented with these two juxtaposed methods, they will wisely choose control. 

In my class, I work to find a happy mix between the two. I focus on designing "tasks" that give students control over certain learnings, but are focused enough that I can also appease the rigid timeline in the front of my curricular documents. The tasks do not require a large amount of creativity, but a willingness to see opportunities to step back and hand control to the students. It involves me being willing to guess where students will go, and ask leading questions to push them further. Every topic becomes an opportunity for me to grow as a teacher, because I don't know where students will go, what misconceptions they will challenge me with, and what discoveries they will make. (By the way, I hate using the word "discovery" because it has become the whipping boy in every staff room conversation. For some reason, teachers doubt their students' ability to make connections without explicit instruction.)

The following lesson is an example of viewing the classroom slightly differently. The original idea was not mine, but as one of my Education professors once told me, "Teachers make excellent pirates." The original lesson plan came from Great Maths Teaching Ideas. They often tweet lessons with interesting connections to nature, sports, and society.

The original lesson was framed under the framework of an "unusual way to teach plotting straight line graphs" by examining the linear function of cricket chirps and temperature. Other that that creative context, there was little difference between this problem and those found in textbooks. They created a worksheet where students were given a function, a table of values, and a grid to plot on. They were then asked a series of questions. A creative, real-world situation doesn't necessarily constitute a change in teacher thinking. 

I took the context and blossomed its potential to include a variety of pathways and afford opportunities for students to practice multiple skills in a cooperative setting. Because of the close ties to science, I chose to place the problem in a lab setting. For background on mathematics labs see "Merit To Mathematics Labs"--an earlier post on this blog. 

Linear Functions Lab

Students are separated into groups of two or three and handed out two handouts. The first is a copy of the "lab report" that looks very similar to the handout from the original lesson, and the second is a copy of a graph that converts degrees Fahrenheit to degrees Celsius. The image I used is below:
They are not given the function which creates this graph for good reason. It only provides another opportunity for a students to create the function once they work with function notation. 

Once all students are settled, I call their attention to the screen at the front of the room and play the following video clip from the hit show "The Big Bang Theory". It provides an engaging start to the lab.

When the clip is done, I show the wikipedia page for the Snowy Tree Cricket. The bottom of the page provides the linear relationship between the "chirps every 15 seconds" and "temperature in Fahrenheit".  Students choose variables, label axes, and then we create the linear function as a group. suggestions are taken and interpreted until we land that if c = "Chirps per 15 seconds" and F = Temperature (Degrees F) then:

F = c + 37

This is a fairly simple relationship, and students are set out to graphing it with a table of values.  Students have had experience with graphing functions before. I insert this lab directly after I have introduced the idea of slope, but before the idea of "y-intercept". As students work, I circulate and ask questions like:

What's the rate of change?
Should the point be connected? (this is an interesting one)
What is the Domain and Range of the function?
What's the temperature if the cricket isn't chirping? Can you know for sure?

When students become comfortable with the first graph, I initiate new learning by preying on their unfamiliarity of the Fahrenheit system. I ask them to switch from Fahrenheit to Celsius to make the graph more relevant. The only tool they are given is the conversion graph--they are not given a function. They are left to devise their own action plan. Should the scale be changed? Maybe the graph is exactly the same? Do we need a new function? How can we make one? What are our new variables? Which axis should represent each?

This second part of the lab provides stratification for students that need it. Some students may struggle with the first plotting, and it gives extra time for them to learn it while providing an extension for those upper level students. It practices reading graphs and interpreting their results. Students will come out with two straight line graphs with unique slopes and y-intercepts. At this point, I ask the students to find the slope of the new graph and then compare them. Which temperature increases faster as the cricket chirps more? We then examine the "b" value in the equation. How does the constant translate onto the graph? We look back at their table of inputs to see what is going on. I might choose to grab a blank piece of graph paper and draw a random line on it. Can we follow the pattern to write a function to represent the new line? 

If students really excel, this a chance to introduce composition of functions. The teacher must be prepared to move along with groups at their own, individual pace. Some may rocket through the graphing but get stuck with the conversion graph. Others may need assistance with basic algebra in the first table of values. 

Assessment is two-fold. Students are required to hand in their lab sheet and graph stapled together with their names on it. Also, they fill out a quick self and peer assessment on how their classmates contributed. Most of the assessment are not surprising because I can gauge the feeling as I circulate from group to group. I make sure that I leave 10 minutes at the end of class. Five is used to de-brief the many learning styles that I saw around the class. I detail various strategies so students get the sense of the diversity. It also highlights hard work and increases motivation. The last five is spent on assessment and other logistical efforts such as hand-in and final questions. 

The context of the problem is creatively posed, but that is not what makes it full of rich learning. It takes a quick shift in teacher perspective to get the most out of creative ideas. Their task is pointed but autonomous. Here students work to expand on prior skills and create new understanding. We are working toward the connection between slope and y-intercept of a graph and the values of "m" and "b" in y=mx+b. A class like this creates an anchor lesson where I can always look back on and say, "remember the crickets". Students dive in and create their own understanding through active learning. You don't have to be the most creative teacher in the world to allow students to proceed on their own, you just need a little foresight as to where they may go. 


Saturday, November 5, 2011

Khan's Place in Math Education

It seems that every educational blogger has voiced an opinion on the growing popularity of the Khan Academy. I am actually quite surprised that Musing Mathematically has largely avoided the topic during its meager 5 month existence. The movement of online lecture snippets has polarized those in the educational community; some teachers detest that Khan claims that sitting in front of his computer can even be close to "education" while others realize the efficiency of his method and subscribe wholeheartedly. I have been sitting passively over the last few months reading developments and arguments, and yesterday evening found an article that solidified my opinion of Khan. As an educator, I applaud his vision and initiative, but I feel like he is overestimating his project's niche of influence. 

The article, which was discovered circulating Twitter several times in an hour, can be found below. It details a sizable grant for the furtherance of Khan Academy. Take the time, if you wish, to read, but I will highlight the telling parts of the article. The problem is not that Khan is going physical, it is that he is claiming to coin a methodology that has existed for years. 

The bravado that angers teachers so much exists directly in the title; to claim that a database of online videos is a "reinvention" is very far from the truth. If we ignore this egregious overstepping of boundaries right off the hop, we get an interesting insight into how Sal Khan himself views his work. He has the voice of a reformer, but overestimates how "reformational" his ideas and methods really are. The following quotations are taken from the article:

"The school of the future will not resemble the school of today," said Salman Khan. "In the past, the assembly-line, lecture-homework-exam model existed because that's what was possible in the no-tech and low-tech classrooms of their day."
The Khan Academy model allows teachers to discover which students are struggling with which concepts, and allows students to repeat sections of videos or online tests until they master the material. One of the goals is to re-engage students, some with significant gaps in their knowledge, who have previously felt lost and disengaged.
"We can now build a new reality, using today's technologies, where learning is custom-tailored and collaborative, bite-sized and iterative," said Salman Khan. "When students learn at their own pace, and become more self-directed, they remain engaged. This helps teachers build strong foundations, so that even students that are labeled as 'slow or remedial' become advanced in a matter of months."
Read the three sections again; the messages are so paradoxical that it takes a while to digest. The text contains two direct quotations from Mr. Khan, and a paragraph of commentary on his invention. The revolutionary language comes through.

"The school of the future will not resemble the school of today"

I guess this statement is correct. Coming from the viewpoint of a mathematician, if we take a long enough look into the future, there may be no resemblance. This is a ridiculous way of looking at reform. Teachers mock statements like these because the method of educating has not been altered by the videos. There is still a single beacon of wisdom and it is transmitting the facts and procedures to those listening. The message is the same, but the medium has changed. 

Khan sets himself up for ridicule with statements like, "We can now build a new reality", because those trained in education see the videos as electronic lectures--a method that seriously lacks revolutionary vibes. 

The part that troubles me is when the academy accredits the "assembly-line, lecture-homework-exam model" to the past. These methods are still used by the vast majority of mathematics teachers worldwide. They are not a phenomenon of the past, but a reality of the present. The irony ensues when Khan himself claims that his new reality is "custom-tailored and collaborative, bite-sized and iterative". His model could not be more overestimated. To claim that 'customization' is as easy as choosing which lectures to hear and 'collaboration' is achieved through stopping and starting a set of instructional videos is ludicrous. He wants to rid the world of the archaic "assembly-line, lecture-homework-exam" model, but claims that his revolution will be built on the back of "bite-sized" and "iterative" pieces. That model sounds a lot like the old lecture-assessment model we have--only on a smaller scale. I find it baffling that a man--who apparently is leading the way--cannot find the contradiction in his thinking. Allowing students to work on topics until they attain concepts is a positive thing, but assessment methods have existed for years to focus on this. Mr. Kahn is right that technology can change education, but has failed to create a method for which that can be achieved. Breaking topics into iterative pieces is not a "reinvention" it is simply a "reorganization".

Amidst these critiques that have been heard the world over, I think there are two large benefits that the Khan Academy can have on education. Each of these can have an effect on educational reform, but does not grab the limelight that Khan is billing in his keynotes. 

1) The Khan Academy can inspire the furtherance of true mathematical reform
I think teachers are insulted that Khan is insinuating that teaching only occurs in the classroom. Teachers hate the fact that a man who has never struggled with students' complex needs is claiming to have the answers. The world of education is constantly changing, and the subsection of math education has had one of the most turbulent pasts. The Khan academy's claims should inspire classroom teachers to break the model that Sal is proposing. Re-invent their practice in a way that cannot be captured in a series of lecture videos. The areas of Project Based Learning, Flipped Classroom, and Inquiry Mathematics are all examples of how teachers are working so the schools of the future do not resemble those of today. 

Numerous sources are developing tools that can effectively harness collaborative and individualistic learning. I have personally posted some of my favourites throughout this blog. Hundreds of teachers gather in the spirit of collaboration on a weekly basis for #mathchat. The Khan Academy should serve as a lightning rod for mathematical reform; teachers need to show Mr. Khan that the schools of tomorrow do integrate technology, but not as a series of iterative videos. 

2) The Khan Academy can serve as a support for the logistical problems of school
I cannot sit back and pretend that the work of Sal Khan is all bad; such generalizations make about as much sense as those made by him. The writer of the article in question captures the true value of the site; it can re-engage students who have experienced large gaps in the past. The idea was initially set up as a way to tutor Sal's cousin over the internet, and it does tutor very well. I have no problem sending students and parents to the site to get caught up. Parents find it empowering to be able to help their children with math that they have forgotten over the years. I do not think that there is any denying the fact that the site is effective as a diagnostic tool for teachers. All teachers attempt to "discover which students are struggling with which concepts", and this a tremendous tool to do so. The only problem is that Khan has over stepped his realm of effectiveness and, in the process, stepped on many teachers' and policy makers' toes. The Khan Academy is a great support resource for teachers. The technology is convenient, but not revolutionary. 

My attempt is to be as transparent as possible. I think that the initiative has a definite place in math education--as a support resource. Maybe the physical centre that is in the works will effectively combine realms of math reform and push the online lectures to the support role they fulfill. The academy should spurn educators forward and provide excellent support for learners across the globe. It is not a "reinvention". In short, Sal has simply misdiagnosed a valuable tool