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.  


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. 

The Task

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.


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 ( 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.