Wednesday, December 17, 2014

Faith, love, and a long story

Thanks to my wonderful audiobook subscription, I recently "read" the excellent Ben Franklin biography by Walter Isaacson.  In the early parts of the book, Isaacson dives deep into Franklin's religious beliefs.  Ben has an interesting faith, as he has Puritan roots but is a product of an Enlightenment Era perspective that truth is revealed through the logical study of nature, not through an involved God (many would call it Deism).  Add in Franklin's practical nature and you get someone who believes that people should be "Christians" who believe in working hard, bettering his fellow man, and living a moral life.  Since Ben Franklin is a person I deeply respect and try to identify with, I found it interesting to hear the story of him figuring out and writing about his beliefs.

While sitting in adoration a while back, I was thinking Franklin, a recent sermon series on "Being a good agnostic" (which is actually about a reasoned approach to understanding and placing faith in God/Christ), and our decision to Baptize our baby Josie that upcoming Sunday.  I was reflecting on my own path in faith and how I ended up as faithful Catholic sitting in Eucharistic adoration about to Baptize the baby in my arms at our Catholic church while married to a faithful Evangelical Christian and deeply connected into a small group at our Evangelical church.  Yeah, it's weird not only attending but being active in two churches.  It can be frustrating, it can bring tears, it takes up time, and those holding extreme views from both of these sides of Christianity might even say it's unacceptable.  Yet, it's what brought me to the depth of faith I'm at and its the engine of growth that continues to strengthen my faith.  I hope and pray that my wife Kirsten and I wake up one day and stop having differing beliefs, and in particular, as believers in objective truth, wake up believing the same right things, but I find this unlikely to happen anytime soon.

Growing up, I was a very open and accepting person.  I connected well with very different groups of people and tried not to offend or exclude anyone.  This perspective crept into my religious views as well, as it has to many in our generation.  I grew up Catholic, as have my parents and grandparents and relatives as far back as I know about.  As a result, we didn't really talk about reasons why we were Catholic instead of some other kind of Christian -- it just was who we were.  To be clear, I strongly believed there was a God, and that he sent his son to Earth who taught us how to pray and live, and that we need to respect and worship Him and do what he told us to do.  Yet, as a selectively rebellious child, the fact that we happened to draw the Catholic straw from the Christian jar didn't convince me that my denomination was any more true than anyone else's.  By age 17, around the time of my Confirmation, I honestly didn't see strong distinguishing beliefs or characteristics between the different denominations and told my parents that I wouldn't be surprised if I switched Christian denominations someday.

In my senior year of high school, I finally mustered up the guts to ask out my first date, the girl who is now my wife (I was just being picky so I didn't have to deal with a failed dating relationship!).  Kirsten was an active Christian at her Evangelical church.  I went to my church each week, and I prayed on my own, but there wasn't nearly the same depth as my new girlfriend had.  The difference in our churches was always an interesting area of discussion for us, but I didn't realize how this big difference would be my core motivator in understanding and loving God better.

Mid-way through the summer leading up to that junior year, while I was working on the Alight Learning software I co-designed with my friends who took a year away from school with me in 2008-09, I flew back home to attend Kirsten's brother's wedding.  Something about the event really hit me a couple days after as I was flying back to Boston -- that could be me.  I had been dating Kirsten for over three years at this point (and had known her well since second grade) and needed to get serious about whether this was going to work or not.  After a little thought, I realized that there was only one potential deal-breaker -- our differing beliefs.  Before I would give up the faith of my family, I had to be darned sure it was equivalent or less true than the teachings at evangelical churches.  This was the proverbial fire under my butt that motivated my deep learning about Christianity and Catholicism.

Leading up to that year, a friend asked me to be his roommate.  I really enjoyed being around Erik, but I was a bit surprised he asked me to live with him as we didn't necessary hang out a ton outside of group settings (though, to be fair, we were in a LOT of the same groups due to a lot of overlapping interests and friends).  Erik was one of the most amazing people I knew, and also one of the most amazing Catholics I knew, a handy coincidence going into a year with one of the biggest decisions of my life.

As for my learning, I started with Mere Christianity (C.S. Lewis).  He begins from a non-believer's perspective and builds up an argument for faith in Christ.  The biggest thing this book did for me was push me off the fence -- Jesus was not a nice guy with nice teachings.  He is either (1) who he claims, the only son of the only God whom we must place our trust in and obey, or (2) a lunatic who has defrauded both the people of his time and every generation since.  It also helped me better understand God's grace given to us at the death of Christ (something I had always known intellectually but didn't really internalize).

Once school started, the group of Catholics that walked over to Sunday Mass together each week now had a new avenue of growth -- a small group.  My friend Jeff (who happened to make up his mind to go to seminary to become a priest around this time) led this effort and we held our weekly discussion in my and Erik's dorm room.  These discussions got me thinking much more deeply about the readings each Sunday Mass, but more importantly, taught me a lot more about the Catholic faith and why we believe what we do.  Jeff and Erik both have an insane knowledge about the Church that I constantly tapped into as we met and discussed.

Along with the group time, the before-bed chats with Erik were the deepest council I received.  He pointed me to apologetics books, helped me think through some of my viewpoints that didn't make sense, and just gave me practical advice when I needed it.

Once we were a semester into school, all of my discussions, reading, and reflection led me to believe that the Catholic Church held the truest teachings of Christ.  This left me a few options: (1) convince Kirsten, using the same information and thoughts that convinced me, to become Catholic, (2) discover that we can be married and use our differences more as a point of growth than division in our life together, and if we're lucky have one of us have a change of heart someday, or (3) break off the relationship since marriage would be impossible.  I tried option (1) for a couple months, but Kirsten had too many deep-seated beliefs that clashed with some of my foundational beliefs.  That left me doing a lot of research on interdenominational marriages.  I was somewhat surprised to find that the Catholic Church was reasonably open to the practice, at least a lot more than Catholic culture would suggest.

When I flew home for spring break that year, I made up my mind and told my parents shortly after walking in the house my intention to get married.  Since I was home for 48 hours, my mom and I found all of my old savings bonds from grandparents over the years and we went ring shopping.  That night I went to my future in-laws house to ask permission to take their daughter in marriage (and thankfully heard a yes!).  After watching Kirsten's Amherst College hockey team win the Division III Championship at the Frozen Four tournament in Mankato, MN, I pocketed the ring and flew back to Olin.  I planned to propose out in the beautiful hills near Amherst that next weekend, but Kirsten was in an unusually bad mood, so I didn't push my luck and waited another week for Easter.  Out on a walk in the woods, I dropped to one knee and she said yes!

The first real shared decision-making about faith took place as we prepared for marriage.  I asked that we get married in a Catholic church with the corresponding preparation courses and meetings so that our marriage would be recognized as a sacrament, something that was important to me.  From there, we asked to take out communion (since over half of the guests were not Catholic and many were used to open communion) and added in the performance of a musical composition written and performed by Kirsten's uncle and cousin.  The wedding itself was one of the best experiences of our lives -- we had an amazing day together with so many friends and family members.

Once married, we immediately (2 days later) were in Rochester, MN participating in our graduate programs (education summer classes for me, physician assistant rotations for Kirsten).  We found two churches that we liked -- a downtown Catholic church and a Christian Missionary Alliance-affiliated Evangelical church in the northwest part of town (Christ Community Church).  We found that, now married, we didn't have the excuse of just attending the local person's church (I went to her church when visiting her in Amherst and vise versa), so we went to two services each week.  After a while, it didn't really feel like a burden -- it was just what we did.  We didn't lay deep roots in either church since we planned to be out of Rochester by the following spring, off to some new place where we would both seek jobs (probably Boston).  Thanks to the incredible draw I felt to my team of math teachers at Byron High School and the good fortune of a new math position being created that year, I had a chance to interview and get hired as full-time staff.  Around the same time, my wife was in one of her favorite rotations in a department at Mayo Clinic that had openings.  She too was able to get a job, so we looked to set down at least medium-term roots and buy a home in Rochester.

As we settled in, we signed up to join a small group of married couples at Christ Community Church.  This group has been amazing and led to the friendships that we treasure most here in this new place.  More than just social, we have done a number of book studies that have continued to develop and improve my knowledge and faith.  In the past year, my relationships with these couples, and especially with the men, have led to excellent discussion, reflection, and growth.

Also around the time we moved, we found ourselves at a 6pm Sunday Mass, a service where priests had been taking turns on rotation from the community churches.  We really liked the priest who had led Mass that night and were curious where his home church was so we could see listen to him more often...and his parish turned out to be less than 2 miles from our house in a building we drove past every day but didn't notice was a church!  We started attending regularly and soon joined as members.  I have gotten involved with a few things, mostly related to youth ministry, and absolutely love the parish, its leaders, and its congregation.

After 3.5 years of marriage and 5 months of being parents, we're optimistic as we continue growing in our faith.  Both Kirsten and I feel deeply connected in both churches, both socially and spiritually.  We are incredibly thankful for the priests and pastors, great music leadership, and the social and volunteer opportunities available to us that all continue to help us grow.  We will need this as we continue to get confronted with some of the fundamental differences in our beliefs in sermons and books, and as we look ahead to more tough decisions such as whether our kids should receive the sacraments of the Catholic Church.

I have very different beliefs than Ben Franklin, but I want to emulate his process of figuring out what I believe, how I came to believe it, and how my beliefs shape my ongoing life.  Like Ben, I want to write about it all as it unfolds so I can have a record of my own journey and can bring in those around me to support and critique me on the journey.

Sunday, November 30, 2014

Assessing creativity and critical thinking

Our middle school has two fifth grade teachers who are using project-based and mastery-based learning in a subject-integrated classroom that they are piloting this year.  For a long time, I have been obsessed with these approaches, and when the chance came last year to bring the idea to life at the middle school, I wanted to be a part of it in whatever way I could.  From my own experience developing new curriculum on the fly, I knew the classroom teachers would be completely exhausted just trying to get through the next days, and collecting and analyzing data that tracks the long-term goals would likely take a back seat.  Thus, I became the data guy for the pilot.

Mastery learning and PBL are simply approaches, not ends in themselves.  The biggest objectives of the pilot were to increase critical thinking and creativity amongst students (without sacrificing core content skills).  Unfortunately, these are hard things to measure and there seems to be no "gold-standard" assessment out there.  Of the assessments that do exist, many are locked up and cost a large, ongoing fee.  We wanted something that we could use for years to measure the pilot against the control group of other 5th graders in the district with little to no cost.

As much as possible, we also wanted the assessments to be reasonably objective to score, though we would trade off some precision for a task that required authentic thinking and creativity.  Our goal was to measure class performance (averages and the distribution), not individual performance, which meant that any individual score did not need to be perfectly precise if we had a large enough sample size to average together (thank you Central Limit Theorem).

Before we could measure these 21st century skills, we needed to get beyond the Twitter definitions (no offense Tweeps) and get something concrete.  We started with the 6-12 Critical Thinking Rubric for PBL and the 6-12 Creativity and Innovation Rubric for PBL, both from the Buck Institute.  To further clarify critical thinking and reasoning, I constantly reference this page from and this very brief summary from Robert Ennis.  Creativity is not as studied and written on as critical thinking, but one expert I kept finding my way back to is E. Paul Torrance.  The Torrance Tests of Creativity focus on a student's ability to problem-solve and think divergently, but they get very specific as well.  Though I thought the tests themselves were too canned and non-authentic, the criteria for creativity laid out on the Wikipedia page provide an awesome start to a task rubric.

Armed with some definitions that the team agreed upon, I started my desperate search for authentic tasks that could be assessed.  I was fortunate enough to stumble on the Ennis-Weir critical thinking assessment at 2am one night.  It is a letter from a citizen to the editor of the newspaper making an argument for a change in city policy.  Students are asked to read the letter and respond, paragraph by paragraph, to whether or not the author is exhibiting "good thinking" or not and why.  It is a very clearly defined task, the task does a good job of assessing a student's ability to read and think critically, and there is a reasonably clear rubric that helps you assess each response.   Because it is targeted at an older audience, I modified it from an 8-paragraph to a 5-paragraph letter, added worksheet-style structure, greatly simplified the language, and made the "reason" section partially multiple choice.  Before starting the task, I will read an instruction sheet with the definitions of all of the choices so students have at least some familiarity with the terms.  See the example task I created.  See the rubric used to score the task.

For creativity, I never did find a good starting point online.  Fortunately, I knew quite a bit about one of the tasks designed by E. Paul Torrance thanks to spending grades 7-12 competing in Future Problem Solving (FPS).  The "team global issues" component asks groups of four students to spend a couple months researching a given topic of future significance such as water, nutrition, or cyber security.  On the day of competition, the team is put in a room together with two hours to read and respond to a futuristic scenario related to the topic of study (such as this one about Space).  As a team, you need to generate up to 16 possible challenges you see in the scenario in as many categories as possible.  Next, you choose one challenge that seems most important and write it as your "underlying problem": Since    (statement from scenario)   , how might we    (verb phrase)    in order to    (purpose)    for    (topic, place, time)   .  Continuing from here, you write up to 16 solutions to your underlying problem in as many categories as possible.  To finish, you write 5 relevant criteria to judge your solutions, rank the solutions against each criterion, and elaborate a two-page explanation of your winning solution.  For a painfully detailed guide to assessing an FPS booklet, read this.

I modified the FPS process to make it something any student could do, without more than a few minutes of training, in 30 minutes.  Students are given a 2-sentence description of a new invention.  Then, they need to generate up to 7 possible groups of people who might want to use the invention, choose the most important one and give a reason, generate up to 7 possible improvements on the original invention that will specifically benefit this user group and explain how it does so, write 4 criteria for judging improvement ideas, defend why his or her favorite idea is good based on the criteria, and elaborate on the favorite idea.  This simplification cuts out a lot of the beauty in the FPS process, but I think it serves as a simple and powerful diagnostic of divergent and reason-based convergent thinking.  See the example task I created.  See the rubric used to score the task.

For both assessments, I made a complete example version.  Note that these examples are not used with students -- the format is identical, but the exact content of the letter or invention is changed to prevent advanced preparation in the topic.  The intention of this is to allow for public critique of the process and make the test and rubric as open-source as possible.  Just as it once was in software, open source assessment is against the grain of common practice.  People worry that if they give away too much detail about the test, that people will just teach to the test, but this is what I want!  If the test is very well designed, and teachers teach to the test, then students will learn critical thinking and creativity skills.  They will buy into the assessment as a valid measure of what they are trying to improve and will use the results to change their practice when it is not working.  I want to design something that impacts instruction, not something that ends up in a buried Excel file.  By opening up the design, I also get to tap into the thoughts of the rest of the world and get critiques from people who know a lot more about this than I do.  It strengthens the assessment and validates its accuracy.  If other people want to start using it or derivatives, it could also lead to a larger pool of comparative data, improved scoring rubrics, and a community around free K-12 assessments of creativity and critical thinking.  Keeping the design of the test cloaked in mystery offers so little upside for everything it misses out on.

Finally, my charge to you: please comment, critique, and question both the tasks I made and the process I used to develop them.  If you are also interested in this kind of assessment, please reach out and let's work together to further refine these tasks and rubrics to make them truly awesome.  I will admit that I only have a couple days to make changes as I will start administering the pre-assessments later this week to the 5th graders.  However, long-term I am more interested in an improved version that is a better measure of student thinking and creativity and can better guide teachers who care about helping their students improve these skills.

Saturday, September 20, 2014

I'm still here

In the summer, I always forget how insanely busy the school year gets, and how quickly it kicks into gear.  Even now, I'm keeping myself up later than I normally would because I know that blogging (at least once a month) is so important for my reflection with so much going on.

This is my first school year with a baby.  Being a dad in the summer has been awesome -- I just love it and I love my daughter.  Now that reality is back, there is suddenly competition between school, family, and self (things like sleeping and taking a true break at lunch).  Thus far, I like the first two too much, so self hasn't been winning, as I spent the second week of school sick.  I didn't think I was sick enough to stay home (except for the end of the day last Friday where swallowing became incredibly painful with my sore throat and I went home).  That said, I forced myself to get 8-10 hours of sleep / night most of that week and actually recovered, so I actually rebalanced in a time of need.  My wife, a physician assistant, works an irregular schedule with mostly 12-hour shifts.  With lots of work days scheduled in September, it has been hard not getting to see her as much until late at night, but that should improve when she goes to 80% time in October.

The other weird thing this month has been living with my parents or in-laws who generously have driven 4.5 hours across Wisconsin to stay at our house and babysit Josie multiple times.  They have been so helpful with both baby support and helping around the house, so it will be interesting to see what things are like when we are more reliant on daycare and on our own.  Despite all of the help, it still goofs up your rhythm when there are other people living with you, so it might take a while to establish a regular routine.  It also will be interesting to see what happens to the house and yard without family around as we try to survive the fall.  Thankfully grass stops growing so much around this time of year...

I was taking Justin Lanier's Math Is Personal online course as we ended the summer.  I feel awful for committing to it when an end of summer trip and the start of workshop week at school completely stole my time away and led to an early dropping-off.  Twitter has been mostly abandoned except for responding to at-tweets and posting out an occasional thought with my free hand while bottle feeding Josie.  I haven't read any blogs since the year started until tonight, and even now only made it through most of my must-reads.  I personally haven't written a post in about a month until now.  I know my participation in this community is incredibly valuable for me, yet it has taken a major backseat.  My only wins here have been already implementing a few new ideas for Stats from Paula and Anthony with my eye always open for more.

Stats has been my metaphorical baby since I took it from my mentor teacher Rob during my student teaching 3 years ago.  This year, Rob was willing to try teaching a section of Stats with the new curriculum I've been (and still am) redesigning.  This is the first time someone has ever taught with curriculum of my design without me in the classroom, so it is both incredibly exciting and a bit terrifying (I don't exactly design for exportability like a textbook company).  We chat almost every day at lunch between my 2nd block class and his 3BC block class about how things are going, how we explain different concepts, and the big ideas behind things.  We share resources constantly.  Rob yells at me when I make up new things the night before and forget to mention it to him.  It's great.  That said, I'm most excited for the end of the semester when Rob (an excellent reflector) can look back with me and discuss the big ideas of the course and how effective or ineffective the design was in teaching those ideas.

In the first unit, I used almost all recycled material from the past two semesters (this is a highly unusual must finally be getting fairly good!).  However, I ripped out two sections that got moved to other parts of the course and replaced them with a project to collect data and design an infographic to tell the story of their results.  Most groups were out collecting their data on their hybrid (work from home) day Friday or on their own time over the weekend, so we get into the meat of the project work Monday.  I intend to show no mercy to bad posters -- my grading reflects a growth mindset of revision, so somehow I feel like I have a license to give feedback like they're college design students.  Hopefully the groups understand that I can have high expectations, they can learn a lot, AND they can earn a strong grade for their efforts, even if it takes a few iterations.

I am also part-way into designing a completely new unit sketched in an earlier post.  The plan is still to teach hypothesis testing and culminate in a debate for the best method to do statistical decision making.  One of my math co-workers, Jen, also happens to be an English teacher and formally teaches speech and debate, so I will continue to bug her as I structure the details of this.  She has already suggested breaking the debate into a few smaller debates, such as a day for the "best way to teach hypothesis testing" and a day for "the best way to do hypothesis testing in practice", since they may have different opinions.  As for the content, I am going to start with a basic introduction to hypothesis testing using StatKey for all calculations (randomization test) to keep it simple.  Then I will teach normal curve calculations, the central limit theorem, confidence interval calculations via the normal curve (and its associated assumptions), and finally p-value calculations via the normal curve.  This sandwiches the material in the big idea while providing enough scaffolding to get into inference calculations with the normal curve as a model.  I'm feeling optimistic about it, except that it needs to be done in a week and I hardly have homework or videos yet.  Don't expect another blog post for a while :(

This class has been a bit of a surprise for me -- I didn't think I would get so into it.  Both Jen and Rob have sections during other periods of the day and we're constantly dialoging about the class.  Rob has been experimenting with a mastery approach to quizzes while maintaining a single-shot test as we've always done.  Jen has been pouring her efforts into applications, examples of Geometry outside the classroom, and interactive activities.  I've been investing my time into content simplification, boiling the material to its essential elements for the most straightforward explanation.  We finished our first unit test on Wednesday.  Rob and I in particular have been frustrated that our results have been so poor given all of the effort we have invested, but we both had great discussions with our classes and we're both continuing generally down the same paths we have been on.  My students had a few ideas, some conflicting, but the one that stood out the most to me was a request for harder homework.  My big goal for the upcoming sections is to pick out a small number of very challenging problems that students can work on together in class.  I think my next step to implementing these well is to get more whiteboards around the room.  I teach in Jen's classroom, but I think she would be on-board if I showed up with a big stack of pre-cut tile-board Monday.

As a team, we have also been changing our common assessments to make them shorter, more focused on essentials, and clearer to students.  We also clarified a policy on bringing notes into the test that is slightly more liberal than I previously interpreted our policy.  The purpose is to avoid a focus on memorization of non-essential elements and practice higher-order application.  The discussions are long and philosophical, but very powerful in deciding what we really care about teaching to our students.

Independent Math:
This is a class where students use the ALEKS software in a computer lab to work through Algebra 2 topics at their own pace.  One of the major evolutions of this class started a couple years ago when we stopped requiring a certain number of topics to be completed each day (5) and switched to a weekly number.  This was then tweaked to have different targets depending on overall progress, since topics were much easier at the beginning than they were at the end.  Eventually, we had the ability to offer work-at-home (hybrid) days to juniors and seniors who showed exceptional progress.  This class is extrinsic motivation at its best, and I hate that aspect, but can't get over how well it works.  Depending on their current total progress, most students have been completing about 35 topics each week, either to avoid extra work time assigned over lunch or to earn days off the following week (it is particularly nice that the class is at the end of the day).  ALEKS usually doesn't let students "complete" something unless they can correctly do problems 2-3 times in a row, and every Monday recent learning is reassessed, so students are held accountable to actually learning the material.  Rob Newshutz, a second-career-teacher-in-training is student teaching with me during this period.  Together, we can tutor many more students and offer occasional direct instruction on common mistakes or concepts.  It has been awesome working with him.

Student teacher:
Speaking of Rob N., it has been an interesting year having a student teacher.  It seemed strange that I was going to teach someone who is my dad's age, but I realized quickly that it isn't what I do.  Instead, I get to co-reflect each day with someone else who is trying to figure out how to teach math to young people.  I ask Rob all kinds of questions, and then I walk away and ask myself the same questions.  I realize that I don't have answers for most of them.  My big question about everything is WHY.  WHY are you doing that?  WHY does this added value to student learning, and how do you know?  It helps me get back in the rapid-learning mode that I was in when I was a student teacher which has been awesome.  Also Rob happens to know a frightening amount about math and software engineering, so I get to learn things every day on the content side just through conversation.  I've learned so much in three weeks already.

5th grade PBL pilot:
Over a year ago, I went to a seminar on split screen innovation with a few coworkers.  The idea was to have classrooms, or entire schools, in a district with completely novel approaches to teaching so that there was constant, deep, internal innovation in teaching and learning.  I was super excited and wanted to do anything I could to bring cross-disciplinary project based learning into our district.  Thanks to a grant opportunity with the state, our district started a pilot in 5th grade where two interested teachers switched grades and started a fully project-based curriculum together.  I have been able to be involved with some of the planning early on, but during the year, I wanted to stay involved without taking time away from my own classes or my family.  I requested to make working with assessment a duty assignment for me for the year (also piloting a different approach to duty assignment).  Though my bosses didn't consider it the ideal solution, I was granted my request for the year.  I am now working with my student assistant during my prep period to create mastery-oriented assessments that will be connected to the MasteryConnect software.  Students will be able to take short quizzes on their iPads and know roughly where they are at with material.  Teachers can also fill out simple rubrics and pour the results directly into the system, so it is more than just multiple choice.  We will try rolling out the first attempts with this starting with math this week.

Digital Learning Coach
I loved being a DLC the last two years, but I REALLY love having fewer meetings to attend every month and no paperwork (okay Google Docs work) to account for teacher progress on technology goals.  As much as I miss this job, I appreciate the extra time I now have to focus on my students and my family.

This season, we're getting everything moving much sooner and in a much more focused manner than we have in the past.  When we created the team two seasons ago, I started out mentoring with one other person.  By the end of the season, we picked up a small core of parents who joined us.  Two years later, before the season is even starting, we have eight adult mentors committed to leading along with me.  I am transforming my role away from the direct teaching and supervising of kids to serving our adult mentors and newly elected student leaders (something we didn't really have a year ago) so they can teach and support everyone else.  We have plenty of capable brains and bodies, so I need to make sure everyone is being utilized effectively.

One component of this is our new Leveling / training system.  Each sub-team, led by their adult mentors and elected student leader, will put on training events for missions in their area.  Everyone on the team will receive level 1 and 2 training as soon as possible to make sure everyone is brought into the team culture and knows enough of the basics to jump in and be helpful at any point during the build season.  In the past, too many people didn't understand very much and were either bored, misbehaving, or frustrated that they were unable to participate.  I hope the structure of the sub-teams and the very specific checklist-nature of the missions and levels will help us get past that.

Our biggest struggle is programming.  After lots of early problems with LabView and C++, we are comfortable enough to make the jump to the most common language, Java.  In addition to a new language, we are going to try to adopt the common "Command-based" framework for structuring our code.  Besides a high bar of complexity that makes it hard for students and mentors to learn, it is impossible (as far as I know) to test code without a robotic electrical system to play with (of which we have only one assembled and working and we rely on this one for our main robot).  Thus, as a programmer by training I am working with a couple students to try to figure this whole thing out before our preseason competition in early October.  Making progress...

Well, I guess there is a lot going on.  I'm glad I finally wrote most of it down.  Maybe if I do this in smaller but more frequent chunks I won't have to write a small novel!

Wednesday, August 27, 2014

A different way of looking at Algebra

I accidentally stumbled across this old doc I made a couple years ago.  It breaks down the concepts of algebra into a few different types of categories and suddenly seems more useful than I remember it being when I made it.  I would love some feedback / pushback / thoughts on this way of classifying the subject.  Thanks to Kate @nerdypoo for her monster comment that has helped me improve this by the time you read it -- but more comments and refinement are welcome!

Modeling scenarios:
  • Every known value can be represented with a number
  • Every unknown value can be represented with a variable
  • An expression combines multiple values (2*boys - girls, 3x^2, etc.)
  • An equals sign creates a balance/equivalence between two or more expressions
  • When there is a single variable among numbers in an equation, it is possible to find the numeric value of the variable
  • When there are multiple unknown variables, a relationship between the variables can be created to see many possible solutions

Classifying 2D equations:
  • One variable mono/polynomial; arithmetic growth:
    • linear y = 3x
    • quadratic y = 3x^2, x = 3y^2 -3x + 2
    • higher order polynomial y = x^3 - 3x^2 + 4
  • Inverse of one variable mono/polynomial:
    • square root function y = sqrt(x - 3)
    • nth root function y = 5 root (4-x)
  • Two squared variables in conics:
    • circle x^2 + y^2 = 4
    • ellipse 3x^2 + y^2 = 4
    • hyperbola 3x^2 - y^2 = 4
  • Geometric growth/decay:
    • exponential y = 2^x, y = e^(2x)
    • logarithmic y = log(3x), y = ln(x - 2)
  • Trig:
    • cyclical waves y = sin(x), y = cos(2x)
    • cyclical waves divided y = tan(2x), y = 2cot(3x)
    • inverse of cyclical waves y = csc(x), y = sec(x)
  • Recursive: a0 = 1, a(n) = 3*a(n-1) + 4

Analyzing number types:
  • Real numbers: any number that can be placed on the number line
    • Rational numbers: using fractions to represent parts of numbers
      • Integers: counting with a way to go positive and negative using whole numbers
        • Whole numbers: counting with a way to represent nothing (0,1,2,3,4,5...), and natural numbers (no zero: 1,2,3,4,5...) [US definition]
    • Irrational numbers: numbers that can’t be captured as a fraction and never end
  • Complex numbers: makes 1D numbers become 2D with a real and imaginary component
    • Imaginary numbers: any real number multiplied by the square root of -1, a useful construct for dealing with the common problems caused by negative square roots

Analyzing operations:
  • Addition:
    • moving left/right on a number line
    • with fractions, keep common denominator (factor out the denominator and add the numerators)
    • with vectors/complex numbers, add each component separately (factor out the x-components or i-components and add leftovers)
    • with polynomials/radicals, treat each power of x or each type of radical as a separate component and add each separately (factor out the common sqrt(5) or x^3 and add leftover coefficients)
    • subtraction = addition with second value as a negative
  • Multiplication:
    • finding area on a grid
    • cancel pairs of negatives (a non-obvious rule)
    • think “repeated addition”
    • with fractions, multiply numerators AND denominators separately
    • division = multiplication with second value as an inverse
  • Exponentials:
    • think “repeated multiplication”
    • with fractions, raise numerator AND denominator to a power separately as you would when multiplying
    • logarithms = undo an exponential equation to isolate the power term

Equation / function graph transforms [ assuming a form similar to (y-k)/b = (x-h)/a ]
  • Translate:
    • Move right by subtracting directly from x, move left by adding directly to x
    • Move up by subtracting directly from y, move down by adding directly to y
  • Reflect:
    • Reflect over the y=x line by finding the inverse function (switching x and y)
    • Reflect over the y-axis by taking the opposite of all x’s
    • Reflect over the x-axis by taking the opposite of all y’s
    • Reflect over any vertical or horizontal line by doing a translation, then an axis reflection
  • Scale/stretch from origin:
    • Divide one side of equation to stretch out along that axis
  • Rotate:
    • On 90 degree rotations, the x and y coordinates change places and one value becomes negative depending on direction.
    • For 0-89 degree rotations, it may be easiest to change to polar coordinates x=rcos(t) and y=rsin(t), then add or subtract from t.

Forms of equations:
  • Function:
    • solve for y, y = mx + b, y = ax^2 + bx + c
    • purpose -- make it easy to find an output given an input, easy to create x-y table
  • Expose an important point:
    • Point-slope form for linear, y - y1 = m(x - x1)
    • Vertex form for quadratic, y = a(x - x1)^2 + y1
    • Standard form for circles, (x - x1)^2 + (y - y1)^2 = r^2
  • Highlight zeros:
    • factored form for polynomials, y = a(x - r1)(x - r2)
    • purpose -- make it easy to cancel terms in rational expressions, make it easy to find where graph crosses x-axis (useful in well-designed applications)

Using the coordinate plane:
  • Goal: make a problem easier to understand or solve by mapping two different quantities (complex numbers, (x,y) coordinates, latitude/longitude, vectors) onto a visual (requires 2D space)
  • How: notice differences between a single answer (a coordinate), a relationship (an equation), and a region of acceptable solutions (an inequality or set of inequalities)
  • Use for non-equations: map shapes onto a coordinate plane to more easily find distances and angles

Parameterizing scenarios:
  • Functions:
    • take an input (usually 1 variable called x)
    • have restrictions on what the input can be
    • create an output (usually 1 variable called y)
    • the output can be classified by the bounds it fits within
    • accomplishes something specific
  • Domain is the set of input restrictions
  • Range is the set of output limitations
  • End behavior:
    • Purpose: to know which term dictates the long-term behavior and in which direction
    • In rational expressions, the highest power term dominates towards positive and negative infinity
    • In some expressions, there is an asymptotic line that the output approaches

Accumulation and rates of change:

  • Physics:
    • Acceleration is rate that the velocity changes (wrt time)
    • Velocity is the rate that the position changes (wrt time)
    • Position is where something is at a given time
  • Rate of change is found using a tangent line of a function at a given point/time
  • Accumulation over a period of time is found by measuring the area bounded between a curve and the horizontal-axis.
  • Starting with one of the 3 physics graphs, rate of change allows you to move towards acceleration and accumulation allows you to move towards position.