CEP800 Technology Integrated Lesson Reflection

The lesson I chose for this project was about discontinuities in the domain of rational functions.  The goal of the lesson was to have students explore and gain an understanding of what causes the domain to be discontinuous so that they would be able to identify functions where this type of behavior is likely to occur as well as at what value(s) the function is discontinuous.  I choose to use the TI-Nspire graphing calculator in conjunction with this lesson.  The technology is not a necessity for students to understand the concept; however, the goal of including this technology with the lesson is to help students focus their attention on the stated goal as opposed to other extraneous, albeit important in previous units, content.

I beta tested this lesson with my wife as my classes will not be working on this for another week.   Just as I hoped she quickly identified that the graph had strange vertical lines appearing in it, representing the discontinuities.  When I directed her attention to the equation I had her enter into the calculator she also quickly identified that the reason that this was occurring was due to the fact that those x-values caused a division by zero which as she noted “you can’t do”.  I created a new equation for her to look at and asked her to predict what the graph would look like.  Again she picked out the values that would cause division by zero.  For our third exercise I created an equation that was impossible to find the zeros of the denominator by inspection.  As it had been years since she solved a quadratic equation, which is not the case for my students, I showed her how to graph the denominator of the equation and asked her what x values made it equal zero.  She estimated the two values from the graph and I showed her that she was correct by demonstrating that the calculator would calculate them for her.  Overall she maintained a focus on the reason for the graphs strange behavior and was able to identify where it would occur, precisely as the lesson had been designed to do.

The lesson did differ from the way I plan to teach to my students in the sense that they have been working with solving quadratics, and finding zeros with the calculator for several weeks.  In this sense the technology should not provide any sort of distraction from the content (i.e. students struggling to use the technology) which has plagued some of my other attempts at integrating technology into my lessons.  As discussed earlier the primary reason for using the calculator is to allow students to focus on thinking about why the function is discontinuous so it is important that one distraction is not replaced by another.

As I thought about this lesson I also discussed whether students should be allowed to avoid having to use mathematics that they have previously learned with other members in my department.  At first, there was a strong sense that the unit on rational functions provided a fantastic opportunity for students to review concepts from previous units, which is certainly the case.  However, when I noted that students seem to get lost when the problems become so long (in the case of a quadratic denominator in simplified form, roughly a half page of written calculations and equation manipulation just to find two x-values) that they lose track of what they are trying to do, several colleagues agreed that the review of old content in this way may be distracting especially when our real goal is to focus on the new content.  I proposed that we may want to consider asking them to do it without the technology at a later time, once the new concept was more fully developed. 

Another interesting idea that came out of these discussion was that by letting students use the calculator to find the zeros of the denominator they are freed up to explore denominators of higher degree (there is no generalizable algorithm for solving polynomials higher than degree four and the generalized solution to degree three and four polynomials is well beyond the scope of an Algebra 1 class).  Hence, our students would be able to subvert these formulas and still be able to make sense of what their solutions mean, even if it was a degree 50 polynomial and have it be almost no more work than if it were a first degree polynomial.

Given the success this lesson was met with when I piloted it with my wife I believe that the overwhelming majority of students will be able to provide a detailed account of what causes discontinuities and how to determine their values immediately after this lesson.  The question that will be more difficult to answer is regarding what the outcomes will be at the end of the unit.  Students typically have a much more difficult time being thoughtful about discontinuities when they are also responsible for determining the x-intercepts, y- intercept and end behavior of a rational function simultaneously.  One other major goal of this lesson is to make sure enough time is spent focusing on the “why discontinuities exist question” that it is never entirely reduced to algorithm and therefore not easily confused with other such algorithms.

CEP800 Digital Story

A video about my experience with the role of scaffolding in the classroom.

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Student Understanding Interview

Check out my Student Audio Interview!

Wicked Problem Project Part D

Formative: Did the project get implemented as planned?  
There were no major deviations from the original proposed plan.  It was interesting to look at the feedback I had gotten about my plan however.  It seemed clear that I had not explained as clearly as I could have the reasons for and ideas about how to implement my website.  As of right now I have a lot of work to do to get this up and running for next year.  There is a ton of content that needs to be written.  Thankfully I have the help of the teacher that I will be working with next year.

Summative: Evidence of success in addressing the problem of practice.
This is a hard question to answer as this was not intended to go live until the '11-'12 school year.  We plan to enlist the help of some students this summer in gauging their reactions to the work that has already been completed and how it can be bettered.  This is really formative in nature but since I will have no other evidence of success it will have to serve as summative until I can see what really happens.  I can say that my teaching partner for this project was really excited to see the work that I had done and was anxious to add to it.

How would you approach another project of this type differently given what you’ve learned here?
I don't think that I would go about doing much differently that I have.  I believe strongly that this will be a exceptionally useful next year. For my personal sanity, I might have chosen a project with a smaller scope.  It has been really difficult to deal with the vast number of things (different technologies, producing content, aesthetics...) required to produce this project.  However, I think the fact that it was difficult and did force me to think about so many different aspects is what makes it a superior choice to an easier topic.

What are the lessons learned that others might benefit from knowing about?
I think the biggest lesson to be learned is that it is a bad idea to pick a technology and then try to wrap a context around it.  From having watched my group members blog postings it seemed like many of them realized there were some issues with their chosen technology once they reached the TPACK phase of the assignment.  TPACK obviously provided a reflective tool to evaluate whether a proposed solution was good or not.  However, I think TPACK is much better used as a tool to guide me in the construction phase.  I would prefer to see Part C of this assignment exchanged with part B for this reason. 

In what ways will you endeavor to do the same project again, and what will you change or not do?
I have a feeling that I will be busy for some time completing the current project.  I anticipate that it will be continuously revised and refined once we see it in action.  In that sense I am not sure that I will move on to doing another project like this for quite a while. 

Mobile Learning Lab

Alright, I played with Polleverywhere for quite a while.  I could see using it in my classroom as a formative assessment tool for sure.  I liked how quickly it registered responses (I sent several to my own question).  The display as a bar graph for multiple choice questions is very nice.  I have used my TI Navigator system in the same way, but it is cumbersome to set up and get kids logged into.  I am less optimistic about the open response questions.  The data isn't aggregated in a way that is as helpful as it could be.  For example I sent the same response twice because I wanted to see how it would handle the information.  The result was two separate answers that had the same text.  One cool thing about the Navigator system is that if it gets the same response multiple times it shows that in a bar graph by increasing the length of the corresponding bar.  I also don't like that you cant ask multiple questions on one poll.  I see using this as a way to get a quick fix on what students know and don't but prescripting questions might be difficult as I don't always know what direction students will go with a particular topic. Here is one of the polls I made.

Make a live audience poll at Poll Everywhere

On the Classroom 2.0 site people cited other potential issues with using it.  Cell phone reception seemed to be one possible limiting issue as well as those who don't have unlimited text plans.  Also, in reading through the 25 Practical Ideas for Using Mobile Phones in the Classroom there were some other issues raised about students who may not have a cell phone and possibly feeling embarrassed if they can't participate.

I have watched kids take pictures of homework assignments from the board so they didn't have scramble to write it all down.  I have even taken a picture of some students work with my phone so that we could save it to talk about the following day.  I think that there are lots of uses for phones in particular and I think as educators we need to learn how to encourage kids to use them creatively and not in the disruptive "I need to text my friend who's in German class right now" way.  Several of my colleagues refuse to believe that students are capable of doing much other than the former and I keep arguing that the reason for that is that we are not showing them what it means to use them appropriately.  That is a separate issue for now I guess.

Data Visualization Lab

Ok, to be honest I didn't expect much from this lab.  As a math major and physics minor I spent huge chunks of time in college making visualizations to support my work.  This was particularly true in my physics labs.  I decided not to look at visualizations that I was used to working on in those classes and instead looked at some of the flow-charting options that were available. 

I have used MS Visio on a couple of occasions including recently in my Wicked Problem Project and I think it has some great features.  For example if you drag a balloon onto a page with an existing balloon Visio will snap the new object to a grid that aligns it vertically or horizontally with the existing item.  Really this snap feature is useful for drawing arrows between items, crossing out a balloon, etc.  Visio doesn't do a great job with pretty colorful charts.

For this lab I played around with three pieces of free flow-charting software.  Dia (http://live.gnome.org/Diav), Exploreatree (http://www.exploratree.org.uk/) and Gliffy (http://www.gliffy.com/).  Each had their own relative strengths and weaknesses.  I am already thinking that it would be cool to use one to set up a comparison chart.  Here it is, double click to take a closer look..

The winner in terms of functionality is Gliffy.  Presumably that is why it is the only one that you eventually have to pay for.  I actually used it Gliffy to make the chart.  Excel might have been better for this task but still pretty neat.

The real visualization I choose to make is a chart showing the normal/ideal flow of students through our math curriculum. 

Wow, talk about professional looking results in about 40 minutes.  My big ah ha moment with this came when I noticed that as elements are moved the software provides the coordinates of the object.  This allowed me to line them all up neatly.  I later discovered that it was possible to enter the coordinates to locate the object, as opposed to moving them into place pixel by pixel.  Not as nice as Visio's snap feature but very acceptable and better looking results in less time than Visio (at least the antiquated 2002 version that I have).  I will definitely be using this type of thing more often, especially as I continue to put together my website.

Wicked Problem Project

After 8+ weeks I am finally done with this project.  What follows the video is a summary of the problem addressed, information about the proposed solution and its connections to the TPACK framework.

The Problem:

The problem facing our district and likely many others is that students who struggle in their mathematics classes.  Often these students reach a point where their self efficacy becomes so low that they give up, resulting in failing multiple trimester of Algebra 1.  The consequences of this are wide ranging and include students not graduating on time if at all, schools failing to make AYP… clearly in the context of No Child Left Behind the last outcome is unacceptable but more importantly as educators we know that the prospects for these students is bleak.  In an effort to increase student achievement my district has dedicated four hours of instruction time for interventions with them.  The teachers will work with students in some way to promote success in their current math classes.  The additional constraint was added that the class must be technology based.

The Solution:

We will use a math lab that is technology based, with the goal of helping students be more successful in their current math classes.  The lab is scheduled to be staffed by two math teachers and cover 4 of the 6 hours in our day.  The lab consists of a classroom with laptops where students will work on assignments determined by their current math teachers and the lab supervisor. 

We decided that we did not want students assigned to the course.  This had been tried previously and the classes ended up becoming populated by groups of students who actively sought to undermine the purpose of the class.  The proposed system allows us to bring students into the lab as needed by calling them out of classes other than their math class or other required courses.   Unfortunately this means that students will be brought down during their elective periods.  It is certain that there will be teachers of electives who are unhappy about having students pulled out of their classrooms.  This is not a statement that elective classes are less important.  To the contrary, under the current system students who fail end up having reduced choices of electives, as they end up filling their schedule with retakes of required courses.  It is our belief that this will be mitigated by the fact that students passing their math classes should have more slots in their schedule for electives later.  Additionally, we do not intend to pull them from any class for weeks at a time and it is planned that they should be able to continue with the coursework of any elective that they miss.

In order to reach as many students as possible we have decided to construct a website that contains a series of learning modules or groups of content geared toward helping a student with a specific goal.  We will facilitate their learning by monitoring their progress with them directing them to particular resources while at the same time building mentoring relationships.  The modules will do the majority of the work with the mathematics content.  We will do the management work and assist with the mathematics as necessary.

What we will be looking to see is if we are able to lower our failure rates for starters.  This is easy to measure because it is quantitative.  A more difficult thing to measure will be the behaviors’ of the students we work with.  It is hoped that we will have an impact on their attitudes toward school and math in particular.

TPACK:

Technology and Pedagogy:  The main issue here is that our chosen pedagogy requires students to use higher order thinking skills and engage in doing mathematics not just memorizing algorithms and facts.  As such it is imperative that we select a technology that supports this view.  There are various pieces of software that exist to help students pass math courses, however they seem to share the notion that mathematics is about process and memorization.  In order to remain true to a constructivist pedagogical model a technology must be chosen that allows us to author content to address the issues of our learners.  The technology must also be flexible enough to allow different learners to access a vast array of content at different times.  A website is one such technology.  In addition to meeting all of the criteria described above it has the added benefit of allowing many other technologies to be used through it, such as blogs, embedded videos, links to other websites etc.  As excusive authors of our own content we are able to help ensure that the experience that students get is consistent with what is being done in their regular classroom

Technology and Content: What is true about a website is that it allows for multiple other technologies to be used within it.  Instead of discussing the entirety of the problem we can look at how a module may play out.  Lets say for example students are struggling with the notion of a function.  A part of the module devoted to exploring these ideas might be a time lapse video where students are asked to record their ideas about what things the can measure that are changing.  They could also be asked to think about way to show how two of the things that noticed change.  More pointed questions like pick two things that are changing and make a table are also possible.   This same module may contain more directly targeted ideas like a StAIR where the students would be asked to take a table, make a graph from it and explain how the two relate.  After any activity students could be asked to respond to a series of questions about their learning.  Each module would also have an assessment at the end to give us and the student feedback about their progress.  A Google Docs Form could be embedded for this purpose.  A link to a Jing presentation about how to make a rule from a table could also be included.  The technology allows for almost any content topic to be covered in this way.  The format of a website allows for complete flexibility to add many different types of experiences for the student to help them engage in the content more fluently. 

Content and Pedagogy:  Where our program differs most significantly from my perception of other mathematics programs is that we do not teach a skill and then try to teach how it is connected to other ideas.  Our goal is to have students come up with ideas for how to deal with problems that is directly linked to other things that they already know.  They may then develop skills as a result of a deeper understanding of the mathematics, but it is meant to be grounded in understanding, not memorization.  We often provide situations where we expect particular mathematics to be “discovered” as a result but the questions are open and not scaffolded to the point where students are left with no other choice but to do the things we want.  This kind of thinking by students is uncomfortable for many of them and they struggle as a result.  There is a need for students to have extra practice in thinking through this type of problem, as this is what is expected of all students in the classroom.  For some students the extra chance to experience the content from their classes again may aid in their understanding and proficiency. 

We also know that having multiple means to access content is helpful in allowing students to access the mathematics.  Many students may also find that interacting with a computer allows them to move at their own pace through material as opposed to what they may experience in the classroom.  Many students lack self assessment skills which would be a tenant in any of the content that is presented in the modules.   It is hoped that students will be able to see some success with mathematics in an environment that is nonthreatening and not directly tied to a grade.  They can experiment with their ideas and build new ones without having to feel as though others will judge them for their responses.  This is where I take on a very active role with students, encouraging them to work with the content the way they would be expected to in their actual math class.