## Thursday, December 22, 2011

### A Reflection

In the waning moments of my semester, I made the decision to create a "class expert" system to introduce the idea of rational expressions. Designed as an elongated jigsaw, the students were divided into groups and assigned a topic. The connected nature of the ideas made this, in my opinion, the optimal time to attempt this type of framework. The full rationale for the project can be found in the post entitled, "Math Class Experts".

Basically, I can validate my choice on the following factors:
1. Time
2. Student Motivation
3. Teacher Curiosity
The unit ended with a short unit exam; I corrected each exam and cross-referenced the percentage grades to see if those who participated in the experiment could transfer deeper learning into an exam situation. I am not claiming that exams are the be-all and end-all of assessment, but only that they are often the reason why teachers don't try different class formats.

I am very certain that spending time becoming an expert on a topic helped students reason on a deeper level. After presentations were complete, I placed a rational equation on the board. The students had never encountered one before. Slowly, members of different groups pieced a solution path together. A member of the addition and subtraction of rational expressions group suggested common denominators; a member of the simplifying group wanted to cancel the denominator. We began to use our skills in new situations. It was a cool feeling to see the class take over, and become empowered by their newfound skills. The students from the class section which did not give a presentation needed much more prodding through the example.

But what about the test scores?

Students from both classes showed no significant deviations from their average performance in similar assessments. On average, students scored 2% lower on this unit test than on previous tests. This figure was identical in both participants and non-participants. Based on the results and circumstances, I concluded the following:

• Unit exams are not always a good representation of deep understanding (duh)
• Test anxiety played a significant role in determining the scores.
The latter fact was very evident. Some students gave perfect answers to difficult questions during their presentations, but could not answer an identical question on a unit exam. One student explicitly wrote, "I knew everything, but just froze" directly on her exam.
In the future, I would do two major things that I did (and could) not do this time around.
1. I would dedicate more time to allow students to beta test their presentation
2. I would allow students to design their final unit assessment. This could take an interview or oral format
If I were to go from the test alone, this experiment would have to be classified as a failure. As any classroom teacher knows, there are moments of educational excellence that cannot be measured by exams.

NatBanting

## Saturday, December 17, 2011

### Trigonometric Mini Golf

Christmas time brings immense stress for math teachers, at least in my division and province. As the days dwindle away, teachers begin to get a more accurate picture of how much they must cover before semester's end. Once again, I found myself in this position with my Grade 10 Foundations and Pre-calculus class. (Saskatchewan Curriculum) My original plans called for 20 teaching days to adequately cover, in my opinion, the topics of trigonometry and systems of linear equations. Of course, by the time I sat down to calculate this I only had 11 remaining.

In previous years I would have panicked and switched into jam-packed lectures to "cover" all the content. This year I decided to re-think that approach. I wanted to find a project or anchor activity that could facilitate a wide swath of outcomes and motivate a high level of learning so close to holidays. I tried several creations, but settled on this one for its native curiosity and deep flexibility.

The Grade 10 trigonometry unit covers the very basics of right triangle trigonometry. They are required to use Sine, Cosine, and Tangent to calculate unknown lengths and sides in various arrangements of triangles. In some cases, they are also required to integrate the Pythagorean Theorem and the SMAT180 Rule (Coined by a former teacher in my division; "Sum of the Measures of Angles in a Triangle is 180 Degrees".) Throughout the initial days of the unit, I referred to our "toolkit" which consisted of mathematical tools such as Tangent, Pythagorean Theorem, and SMAT180.

I introduced the ideas of trig as ratio with a short prompt involving similar triangles. We named and worked with the Tangent ratio first. After the introduction of Inverse Tangent, they were presented with this task. That background knowledge was essential to completing the task, and opening pathways for deeper learning along the way. Students were encouraged to employ their full toolkit in any way they deemed legal during the activity.

Students chose a partner and were given 5 things:
• A Golf Hole
• A wet-erase marker (Overhead pen)
• A blank protractor tool
• A ruler
• A napkin or kleenex
The Golf Hole outline was photocopied onto a large sheet of paper and securely covered with an overhead transparency. Each hole contained a unique shape,  a starting point, and a hole (finishing point).

All files ,instructions for the blank protractor, and examples of student work can be downloaded here

Also included is a file containing a page of 6 "blank protractors". These tools were designed to be able to create congruent reflection angles for the ball without giving the students a protractor. I did not want students to have a protractor until they used trigonometry first. Once they had calculated the angles, protractors were available to "check" or "cross-reference" their results.

Students were introduced to the blank protractor and then asked to find a direct path to the hole. Essentially, they were asked to find the initial angle on contact that would create a hole-in-one. A sample was placed on the IWB, and I demonstrated how to use the tools provided. Students were to first find the correct path; water was provided to erase and re-start if the ball missed. When they had found a path, they were to segment their path into right triangles and measure the lengths of each leg. As mentioned earlier, a ruler was provided.

Students were asked to "solve" every right triangle in their pathway. All calculations were done on the sheet provided. Once their initial task was complete, I began to stretch the topic as each individual group was ready for extentions:

• How far did your ball travel?
• What tool did you use?
• If you didn't have a ruler, how could you calculate the distance?
• Is your distance the shortest possible? Can you prove it?
• Is there another tool, besides Tangent, that could calculate the angle?
• What if you measured the hypotenuse and a leg, instead of two legs?
• Which pairs of angles will always be congruent?
• What relationship exists between every triangle on your page?
• Can you find a different route to the hole?
• Will the shortest route always be the one with the fewest bounces?
Different groups will take it in different directions. Some are ready to encounter a new ratio (sine or cosine); some have the eyes to see the similar triangles on the course. Others are satisfied by proving their trig is correct with a protractor or the SMAT 180 principle. The teacher needs to be willing to accommodate the fancies of the student. If they want to "jump" a barrier, it must contain mathematical consequences. During my first facilitation, I discovered that all the triangles created on a course containing only right angles will be similar. Can you prove this? Yet another example of the inextricable link between the teaching and learning mathematics.
The Goals

The task was designed as a platform to move from Tangent into Sine and Cosine. It gave them curiosity-led "toolkit" practice. I define the learning goals as follows:
1. Practice Tangent calculations. Cement the use of inverse tangent to find angles.
2. Play with angles and their reflections. Understand what shifting an angle does to its first, second, third, ... , nth reflection.
3. Infuse an active, project-based element to trigonometry.
4. Encourage students to cross-reference within their tool-kit. Demonstrate that the various mathematical tools yield identical results when employed at correct times. Show mathematics' interconnectedness.
The Results

Students handled the task very well. Rich examples of mathematical thought were very evident. Students were very proud of their final results; two comments in particular stood out for me:
"Hey Mr. Banting, look how mathy this looks! These should be put up on your wall."

"See Mr. Banting, we should do more activities like this; this is how I learn!"

Powerful statements coming from 15 year-olds.

In my attempt to conserve time during the hectic end-of-semester time, I ended up creating some of the richest learning in the entire semester.

NatBanting

## Sunday, December 4, 2011

### Math Class Experts

Last night I was preparing my list of things to do. This has become a typical Saturday night activity for myself. Almost every week, I am commissioned with the task of preparing a new unit for one of my classes. I am a new teacher working with new curriculum. These two realities, coupled with my desire to keep my classes fresh, force me to steadily plan and reflect on past preparations. As I sat down to prepare a pre-calculus unit on rational expressions, I quickly became bored. The weekly drone of preparing a unit plan got me thinking:

If I thought this was boring, what would my students think?

I know exactly what they would think. I began to muse on different ways to present the topics in order to give my students a well-deserved change of atmosphere. The Pre-calculus curriculum is packed, and we have rocketed through many topics; a refreshing perspective might work wonders for their learning.

I was also approached by an intern in our building; she wanted to observe me teaching. I figured if I had another set of experienced eyes in the room, I may as well set-up something different that we could try. I should be clear that this will not only be a learning experience for the intern--I will certainly take numerous learnings from the experience as well.

I decided to set up a system of class experts where groups of students (formed by me) are assigned a section (or topic) from the unit. They will be given 3 class periods to master their topic, isolate key learnings, and prepare a presentation to communicate them to peers. I planned the preparation time over a weekend so groups could meet if they so chose. The goal, from my standpoint, is to provide an authentic learning task during one of the hardest times to garner any type of learner engagement--the Christmas season.

In about 2 hours, I had set up a wiki page complete with project explanation, a detailed schedule of the unit, evaluation criteria, and a page for every group to upload their files. This will be my first experience working with a class wiki; I purposely am using it in a limited capacity until I become comfortable with how students react to it.

This experiment has three main purposes:
1. My hope is to show the intern (who will be, and already is, a very good teacher and coach) that mathematics instruction is changing. There are alternatives to the lecture-test-repeat model that is so often beaten to death. I want to demonstrate (hopefully) that crucial learnings can still be attained when students are given some independence. Also, I want to introduce her to the role of math-teacher-as-facilitator-and-questioner. The most important discussions with students aren't about answers, they are about the process.
2. I want to develop my own repertoire of teaching strategies. I find it harder to arrange "open learning tasks" as the complexity of the mathematics increases. My younger students enjoy numeracy prompts and designed projects that allow them to play with concepts and create understandings. I want to get better at teaching complex topics. This experiment fits into my constant professional development.
3. I want to compare the students who participated with those students from an identical class that will be taught in a traditional sense. I have two sections of grade 11 pre-calculus; only one will be using the wiki. By examining the scores on the test and cross-referencing them with previous data, I want to check and see if the class format develops deeper understanding. I must admit, this is huge draw because I am planning a fairly wide-scale implementation of Project Based Learning next semester. A closer documentation of that can be found here (http://musingmathematically.blogspot.com/2011/10/proper-workspace-for-workplace.html) The vision has since changed and I will post results throughout that process.
I have heard (through twitter) from others who have tried something similar; if you have any advice or results from your personal exploits, it is more than welcome. I hope to report great things in the weeks to come. I believe that educators need to take risks to stay in touch with their students; sometimes this results in failure. The ultimate success comes when we take the failures, examine them critically, and continue forward with conviction.

NatBanting

## 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. (http://musingmathematically.blogspot.com/2011/10/proper-workspace-for-workplace.html) 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.

NatBanting

## 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 www.polleverywhere.com. 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.
polleverywhere.com 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.

NatBanting

## 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.

NatBanting

## 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

NatBanting

## Sunday, October 30, 2011

### Destroying Functions

I have spent the better part of 2 weeks going over various mathematical relationships in my Grade 10 class. They have been represented as tables of values, arrow diagrams, and sets of ordered pairs. Relationships, both qualitative and quantitative, have been defined, analyzed, and graphed. My focus on graphical literacy has been previously detailed on the blog. See the link below for details:

Numerous relationships were handled. Students we required to create a family tree and then represent its branches as a table of values and set of ordered pairs. Throughout the various exercises, the words "input", "output", "domain", and "range" were consistently used. My family tree mapped the connection between a Domain of "Names" to a Range of "Familial Relationship". Some of my ordered pairs then became:

(Ben, Brother)
(Kevin, Uncle)
(Marla, Cousin)
...

Qualitative relationships eventually gave way to numerical ones. Students spent time looking at discrete data such as temperatures, test scores, and mileage ratings. It was through these exercises that the terms "independent variable" and "dependent variable" became part of the classroom jargon. On the whole, the topic of relations was well received; I knew that the next step was to define a function--a process that has gone poorly for me before.

I have seen several analogies that are designed to explain functions. Each is usually paired with the textbook definition of function-- "a special type of relation where each element in the domain is associated with exactly one element in the range" (Foundations and Pre-calculus 10, Pearson, 2010).

1) The Function Machine
This is the overwhelming favourite of math teachers. In the function machine model, the input value (domain element) is placed in some kind of illustrated hopper of the machine. Different portions of the machine contain different operations and the result is an output value (range element) that drops out of the machine. The idea being that if the output was two different numbers, the machine would no longer be functioning. Teachers then stack the "Multiply by 2" machine against the "Add 1" machine to create the "2x + 1" function machine. The proper output values are then reduced to a matching puzzle of the possible machine parts.

This model may create a visual crutch for the computation of ordered pairs, but does not help students decipher between functions and non-functions. I wanted to use a method that would make the students actively create and modify functions.

2) The Promiscuous Range
I have never used this analogy, but a student brought it to me from another math class in my building. In a function, the domain and range are given personalities. The domain elements are faithful to their mates. They are portrayed as hard-working, blue collar citizens. If a relation is a function, each domain value would never dream of having more than one range mate--that is out of character. The range values, however, are more promiscuous. A function still allows a range element to be paired with more than one domain element. Students find all sorts of off-colour words to describe the range element; in the end, a function has faithful domain elements but possibly promiscuous range ones.

3) The Split Decision / Uncertainty Model
This is the model I chose to employ; I felt it could be used as a diagnostic tool for students when presented with a relation. I presented the definition to the students and we analyzed several relations in tables of values. The input values always lead to output values; each ordered pair is presented as a decision. If a certain domain value presents a split decision or uncertainty in the answer, the relation is not a function. If the entry "2" appears twice in the relation, but one time it is mapped to "3" and the other to "4", the domain entry has created uncertainty. How is one to answer definitely if there are two options? Students took to this immediately.

I created this image to further represent the "decision" model.

Students are asked if the vending machine is "functioning". (You'll have to excuse the pun). It takes a quick second for students to realize that the input "A7" creates uncertainty. Choosing that input may yield 2 possible outputs--there is no way of knowing.

Functions are practiced for a couple days until I feel it is time to introduce the Vertical Line Test. I do not give them the rule, but rather tell them a story...

The first computer game I ever bought was Sim City 2000. It was somewhat of a cultural phenomenon where the task was to create a functioning city. It gave me my first taste of politics and economics. The game was addictive; hours would melt away as I sat in front of the screen debating whether or not to raise taxes on my figurative citizens. Despite its appeal, there came a point in every game where it was no longer fun to fix run-down neighbourhoods. As seen in the screen shot below, there were several disasters that could be used on your own city. Tornados, hurricanes, and even space monsters were possible. I told my students that we were going to take the same mentality into our study of graphing functions.

I brought up a blank Geogebra file on the IWB and created a blank Table of Values on the adjoining white board. I filled in points one at a time while graphing them as I went. After six or seven, I told the class that the monotony was too much; we were going to begin destroying the function. By themselves, each student was required to create an ordered pair that would destroy the function. After about two minutes I elicited the responses from the class.

As options came, I graphed them on the IWB. After about five or six, I stopped and asked one question:

What do all of these destructive points have in common?

Discussion ensued. By the end of it, we had coined the Vertical Line Test. Students tested their new theorem on sets of points and continuous functions. They now had linked the fact that two ordered pairs lying directly on top of one another created a split decision. They had linked the algebraic and graphical tests for "functionality".

Functions are the base of higher mathematics in high school. Grade 11 and Grade 12 courses focus almost exclusively on them. I employed a framework where testing if a relation is a function is an active activity. Creating the space for students to actively create and destroy relationships builds a thorough understanding of the topic at hand.

NatBanting

## Saturday, October 22, 2011

### Graphing Literacy

My school division has been pushing literacy for a few years now. The division priority has filtered its way down into many programs at the school level. As a basic premise, if students are exposed to literate people and perform literate activities, their skills will grow.

As the term is dissected, it seems that every stakeholder can find a way to skew the term to mean that their discipline is a crucial part of being literate. Reading and writing skills are an obvious avenue, but the ideas of technological and social literacy have emerged as important parts of every student's school experience. Riding shotgun to these ideas is the idea of Mathematical literacy--Numeracy.

Most students carry the misconception that mathematics is a unique commodity that is unlike everything else they encounter in schools. Somehow, math educators have managed to make a system created to understand the world seem completely disconnected from it. I believe that the burden of innumeracy is born from this disconnection. Mathematical frameworks can be applied to situations that were otherwise thought of completely innumerate.

Students learn the Cartesian system of rectangular coordinates fairly early in their High School career. By grade 10, it is used exclusively for the study of functions--algebraic functions. A system of relating variables has been completely transformed into one that is only associated with "x" and "y" and some kind of "slope". Before, and throughout, the unit on relations and functions, my students are encouraged to graph their thoughts--give me a graphical representation of their actions, emotions, justifications, etc.

Students see that changing the dependent and independent variables affects the story that the coordinates can tell. A graph can guide student decisions. Students are encouraged to graph my tendencies as a teacher. If we put "Time into Class" on the x-axis and "Chances Mr. Banting will let you use the Bathroom" on the y-axis, there is a relationship that exists graphically. Students bring a unique experience into the situation. It provides a rich discussion on the shape of the data, because every student is an expert in the field. Examples from textbooks often talk about things so distant from their lives; homework littered with distant topics only further distances mathematics from them.

Open up a discussion or critique on the topic. As the teacher, I lobby to fix the graph to make me look "nicer" or "more fair". Maybe we graph a "fair teacher" on the same grid. How far is my line from theirs? What does that represent? What if we switched the variables? What if the variable became "Chances of getting a Drink"? Would the line be lower? Higher? What would each represent?

Students intuitively graph; I open a unit up with this problem, and most jump right on board with strong opinions. They can begin to apply the mathematical relationships because they are familiar with the data. Discussions of continuous and discrete data are easily pulled from an activity like this. We could have two graphs:

1) Chances of going to the Bathroom after a Maple Leafs Loss vs. Time into class
2) Chances of going to the Bathroom after a Maple Leafs Win vs. Time into class

The vertical translation of these side-by-side graphs can be very telling. Soon the class has a "Mr. Banting Mood Graph". What is the Domain? What is the Range? These will begin as qualitative entities. Maybe we apply a scale of 1-10. Do we include fractions? Maybe the data needs to be discrete? Maybe groups break off and graph the situation themselves and them return with a narrative argument for their solution. When the class is ready to move into function notation, a line of best fit could be developed. Could we extrapolate the data if a class was 2 hours instead of 1 hour? How accurate would the data be? How would the graph shift?

All these questions fit into the curriculum, but emerge much spontaneously. Students feel the independence of changing variables and graphs. Even as I am writing, my script has taken on a tone of freedom and exploration. A simple concept, linked to some powerful emotions, creates a rich mathematical discussion.

I give my students a personal graph assignment every year. After a discussion much like the above, they are given a handout with a Cartesian Plane on it (only quadrant 1). They are asked to pick any two variables and graph the relationship. There is a space for a description of their reasoning below. This is, by far, my favourite assignment of the year. Every inside joke and class dynamic comes out in one form or another.

This year, the staff won intramural volleyball. Two students we beat in the finals are in my class. Needless to say, our volleyball skills come up in many conversations. One student graphed the "Skill level of a volleyball player" vs. "How valuable they are to their team". I made an unflattering appearance on the graph--their ordered pair was very generous. I make sure to re-draw the graph in my interpretation. The assignment became a form of graphical satire.

This activity is very free-flowing, and builds great numeracy skills. It re-inserts a mathematical framework into student consciousness. I get students graphing freelance relationships for me for months after the topic has faded or been taken over by slope y-intercept form. Many mathematical implications emerge by simply playing with graphs.

NatBanting

## Saturday, October 8, 2011

### Proper Workspace for Workplace

My province is in the midst of a major overhaul on its curriculum. This puts me in a very interesting situation. I am a new teacher in a large division filled with veteran teachers that all feel as overwhelmed as myself. I can't decide if this is a curse or a blessing; I simply continue to roll with all the punches that curriculum renewal brings. On top of the nuts-and-bolts of each new course (5 of which I teach for the first time this year), the division heaps on division, school, department, and personal learning priorities. To make matters even more confusing, each initiative comes with about 35 acronyms. I can't tell the difference between AFL, PLO, PLP, PPP, SLI, PBL... you get my drift. Amidst the chaos of red tape, I believe I have found something to hang my hat on.

Our department goal is to find creative ways to develop and foster a growth mindset in our students. Our school has a very large proportion of unsuccessful students. Many students feel as though math is too difficult, or genetics has blocked their possible success. This is not a phenomenon specific to Tommy Douglas Collegiate. I read Dr. Carol Dweck's book Mindset last year as a part of school book clubs, and found its message intriguing. I coupled this with research I have been doing into Problem Based Learning in Mathematics. The creation of the new "Workplace and Apprenticeship Mathematics" pathway in my province completed the perfect storm of personal learning. All these factors contributed to my brainchild.

A little more explanation needs to go into the pathways in Saskatchewan. Grade 9 math is taken by every student. In Grade 10, there are 2 options--Foundations and Pre-calculus 10 and Workplace and Apprenticeship 10. In Grade 11, FPC10 splits further into 2 pathways--Foundations 20 and Pre-calculus 20. Workplace and Apprenticeship Mathematics 20 is also offered. In Grade 12, all 3 strands are offered at the 30 level along with Calculus 30. The whole idea is for students to take the mathematics that is suited for them and their future.

The Workplace and Apprenticeship pathway (affectionately coined "A&W Math" by my department head) is designed to build skills pertinent in a work setting. The textbooks are designed around unit projects, and the lessons focus on application. Links to the Saskatchewan Curriculum are found below:

I strongly feel that the spirit of the courses require that they be taught in an authentic setting. This feeling, combined with the department initiative and some work with PBL, has begat my vision for how my 3 sections of A&W math ideally will look next semester. My hope is that the reader of this post will contemplate my thoughts and comment on their feasibility. Any feedback from prior experience or similar experience is greatly appreciated.

I want to set up a completely project based environment for the course. This begins with the set-up of the room. Desks are removed and replaced with small, circular tables with 3-4 chairs around each. I have chosen 3 as my optimum group size, but will leave room for special circumstances. My classroom is equipped with a SMART board and I am working on access to a half-pod of netbooks. Unbridled access to internet resources will only fuel the independent feeling I want in the room.

The format of the course will look roughly as follows:
• Every student/group is given a binder with a print out of the curricular objectives in it. The binder includes rubrics, a list of sample projects, and a stock of daily log sheets.
• Students/groups will design/brainstorm projects that fit specific curricular objectives. They will create a project proposal and get it approved by me before they start.
• Each project works on highlighted areas of the curriculum. Their outcomes will be physically highlighted in their binders.
• Every day ends by filling out a daily log sheet which details who was present, what was discussed, decisions that were made, progress to report, and a plan for the future.
• Every project ends with a product. A short presentation (business style) is made to me or the class at the conclusion.
• Students are able to overlap objectives in successive projects to attain a higher grade for that outcome. The grade for that outcome is never fixed; students may always grow in their understanding of a topic.
• Evaluation of each project is based on a rubric (developed in tandem with the group). Self and group evaluation will also play a major role.
This is an early stab at my vision. I have rough outlines of the group log sheets and rubrics, but nothing is final yet. My hope is to present this framework as my attempt at developing a growth mindset in my students. It also satisfies my curiosity for PBL and creates an authentic workspace for the the workplace mathematics to unfold. There are a few problems that I have to address (and if you can think of more, please comment!)
• What do I do for students that do not show up for class?
• What about students that complete all goals before the semester ends?
• What percentage of the goals completed can be considered a pass?
• Do I simply check off each outcome, or give a letter grade for how well it was met? Possibly a rubric 0-4?
• How can I possibly control the freedom granted with group netbooks?
• Do I allow very similar projects?
• How large of a scope can a single project encompass?
• How should I handle an in-class topic that needs to be widely clarified?
Ironically, the design of this class pathway is a perfect example of Problem Based Learning for myself. My short career has not contained an idea quite like it. It requires tremendous administrative and departmental support (which I feel I could get), but also needs wide-scale fine tuning. I am hoping the twitterverse can lend me some all important critical feedback.

I look forward to your remarks.

NatBanting