Feb 202016
 

As I have been examining my practice through the lens of Critical Theory, I asked myself how would I teach differently now than I did even a year ago? Great question.

If-you-change-the-way__quotes-by-Wayne-Dyer-40  It is time for me to look at AP Stats differently.

The last year I taught AP Statistics, I created great connections through the entire year on each topic, how each piece fit together, and how the end results started from the beginning topics. I carefully planned it so that every element of the year connected. And then, after the AP Exam, we had 4 weeks where I challenged the learners to come up with a question, do the research, and answer the question. Topics ranged from bullying, treatment of gays in stores, to trash on the side of the road. A huge, broad range of topics.

But, we did not do anything about those topics. We didn’t share them with the community. We didn’t have time. We collected great information, but we did not ACT on it.

If I teach it again, the first week is answering the following questions.

  1. What problem in our community do you wish to solve?
  2. Is this problem something on which we can we collect data?
  3. What data do we need to collect before we can formulate a solution?
  4. Share with class.
  5. Are there any similarities in problems?
  6. Can we consolidate any of the ideas?
  7. Discuss.
  8. Revisit 1 – 7 until we can not do 6 any further.
  9. List the topics for the class.
  10. Form groups for each topic based on your own interest and your own passion.

After that first week process is over I would think we have between 2 and 7 different projects in each section. I would have to be flexible and let the class drive the number and type of projects. The only thing I can think of why to reject a project is if we would have to deal with FERPA violations, incredibly sensitive topics like rape or incest, or other legally sensitive issues.

This is the truly difficult part of the teacher’s role, is playing the gatekeeper. I would want the learners to make decisions on what they want to study, but I know that there are some topics that are not researchable by high school learners. We don’t have an IRB to do experiments on people, for example. But we want some groups to do experiments. So I would need a committee of people at the school willing to be the final Yes / No on some topics. This is actually true to the real practice of research.

After we have decided on the specific topics, then we start into the process of answering the following questions:

  1. What types of data are there in your question?
  2. How do we display those types of data?
  3. How do we collect the data in the most scientific manner?
  4. etc, etc, etc.

These are questions that come out of the AP curriculum word for word. The only difference is that I, as the teacher, will be phrasing the lessons in the context of their projects. We will be learning from the different groups why we need to know about categorical and quantitative data. We will be learning from the different groups why bar graphs work for one type, but not for another type, and we will have to dive deeply into cluster, stratified and every other type of sampling in order to come up with the BEST way of collecting data for each project.

The goal now is to dive into the AP Stats curriculum deeply. We won’t need to come up for air because we will be inhaling the vapors of our excitement for our project. (wow, that metaphor was tortured, wasn’t it?)

  • What if a learner wants to switch? I don’t think there is a problem with that. Let them choose their enthusiasm.
  • What if an entire group decides they are more passionate about different other projects? Great, then we dissolve that research group and form a new one.
  • What if they decide to start over with a new question 1/2 way through the year? If they really want to go backwards, and redo all the work they have done on experimental design, research design, question creating, data analysis, and all of the rest of the topics then why stop them from relearning the material in a different context? Granted it is a ton of work, but they are learning, relearning, and taking charge of their education on a topic they are interested in. Why block them artificially?

Second semester is about finishing probability so the learners can moving into confidence intervals and tests. This is where the decision making comes into play, and as the learners become confident in this area, they will be making decisions on their topics.

Those last four weeks of school when I used to do projects would now be turned into “Action.”

  • Meet with the administration or counselors of the school about the data collected and share the statistics and conclusions. Work with the them to come up with a plan to solve problems, or at least come up with a plan to work on solving problems.
  • Write letters to the newspapers and media.
  • Write letters and meet with politicians.

The end goal is to allow the learners to drive the content of the class. They would be much more engaged in their own questions than any question I could come up with.

They would still be learning 100% of the AP Statistics curriculum, but now they would be more engaged and see the purpose for each “module” of the curriculum in a more solid, substantial way. This should help with AP scores (but I have no data to support this).

And in the end, hopefully it would make the community (however the learners bounded this) better.

What it would take from me is a huge willingness to give the learners power over their own education. They would have the ability to make decisions, and be allowed to follow through on those decisions. Some of those decisions will not turn out with positive (statistically speaking) results. They will get negative results. That is real life.

It would take time to plan, to organize content around their projects, and to think deeper about the connections. It would take time to connect with admin and parents to explain why I am doing this. It would require the admin of the school to be willing to allow learners to have the power.

It would absolutely weaken the oppression of the learners done by curriculum designers.

I want to do this. I am not in a classroom any more to do it.

Is anyone willing to partner? I will help. I will support. I will do everything in my power to make your life easier while doing this.

I think it is worth doing.

Feb 192016
 

Learning is funny. There is an entire realm of things to know out there in the real world. Yet until you start looking, it is so easy to gloss over all of those things.  I started looking with a more  critical eye at the world and at my own practices, and realized that I was not meeting my own objectives of inclusiveness and change. This post is an extension of my last post: My Critical Pedagogy / Theory Journey.

Early this week, Adisa Banjoko and Arash Daneshzadeh posted an article named: Why Don’t The Black Kids Like Math and Science?: Easy Answers. I was intrigued. They promised easy answers, and I doubted there were any such things. I suggest everyone read the article. I will have my learning pre-service teachers read it this semester as well.

easy-button

Have you read their article? Go do it. It is worth it. Just promise you will come back.

There are two paragraphs that explained  teaching pedagogy I want to point out.

I explained to one of the math teachers, a White female, how crucial this was. “If your students are mostly Latino then you need to tell them about how the Mayans invented the concept of the zero several hundred years before the people of India and they had no contact.” I talked about how Aztec and Mayan architecture is something that should be used to as a cultural bridge for them to understand their legacy in math and science.

Her vacant eyes she blinked in hollow despair “But I don’t know all that stuff.” Her unwillingness to pursue new racial and cultural paths to math told me she was not interested. She still struggles to keep her students engaged to this day. via (Emphasis mine.)

Ms. Banjoko and Mr. Daneshzadeh give resources that will take any teacher from zero to … well, not a hero but at least a sidekick, pretty quickly. There are so many resources offered in the post that it will take me hours to read them all, but I am in the process of doing just that.

But, think about the teacher in the quote above. “Her unwillingness to pursue a new racial and cultural path…” Ouch. How many teachers are willing to take on the task of teaching content AND teaching culture? That is the problem the White female teacher in the post had. This teacher defined her job as only teaching math.

Her job, my job, is not to teach math, it is to teach PEOPLE. What is the best way to teach people? Well, it depends upon the person, really. That is the point.

The New Teacher Center created this meme several weeks ago. I saved it because it spoke to me.

3 habits

Why? Because of the statements embedded in it. “Seek growth opportunities, take responsibility for learning.” Hmm, the White teacher above did neither. They had an opportunity to grow AND take the responsibility for learning by learning themselves. That is a no brainer.

“Take risks and try new strategies.” Holy Hell Batman, the authors of the article handed the teacher a new strategy laid out in detail. The White teacher above walked away.

Ms. Banjoko and Mr. Daneshzadeh promised easy answers in their post. I am not confident the answers are easy. I think it takes a lot of work to learn about the individuals who are in my classroom. I think that work is mandatory, however. It is part of teaching people. It is part of leading while following, to paraphrase Freire.

If we want to have classroom where we are not subjugating the learners, where we are not oppressing them and holding them back from learning, we need to start celebrating them and giving them back the power that has been stripped away from them.

Reading the multitude of links provided by Ms. Banjoko and Mr. Daneshzadeh is a first step. Learning how that applies to the learners in the room with you? That is a great second.

Feb 142016
 

This is a class assignment. Not to blog about it, but to write a paper about it.  The “it” is critical pedagogy/theory. Do you know what that means? I didn’t (and probably still don’t) either. I realize that I acted in ways that inched towards critical pedagogy, but I didn’t understand the theory. When I say, “I inched towards,” I mean it. I think I was heading in a direction that was taking me towards being a critical educator. I was not there.

quote-we-cannot-all-succeed-when-half-of-us-are-held-back-malala-yousafzai-59-1-0176
When I first read this quote, I thought “Females are the half being held back. Right, I have to teach to all learners, not just the male learners.” But, as I developed ideas of critical pedagogy (again, before I knew what it was) I thought about my learners who didn’t like math. Was I holding them back? What about my Hispanic learners? Or the gay learners? Or what if they are gay, Hispanic, and female!

The next question I started asking myself is, “If I am not teaching to Hispanics or females, am I holding them back? After all, If I am teaching ‘neutrally’ then isn’t their problem to learn, not mine?” I have heard a version of that question from many teachers: I teach, learners learn, that is the way it goes. I once told a teacher, “A teacher who says I taught it, but they didn’t learn it is the same as a salesperson who says I sold it, but they didn’t buy it.” (No, really, I told that to the teacher’s face. They were … not happy with me.)

If I am not teaching TO the female learners, or TO the Hispanic learners or TO the low SES learners, than I AM holding them back. Purposefully, with foreknowledge, and now with malicious intent. I am using my power as an educator to purposefully hold back some learners over others.

I can’t do that. I will not be that teacher who doesn’t teach to ALL my learners, to get ALL of the learners to their maximum potential. I will not be the teacher who abuses their power.

But does that mean I am a critical educator?

The short answer is No.

The long answer is also No.

Because I didn’t create a classroom environment where I challenged the learners to engage and change the world. I changed my classroom for them, but what did I encourage them to change?

That is the difference between being an aware, a reflective, educator and a Critical Educator. And the more I learn, the more I believe every teacher SHOULD be involved in critical theory / pedagogy. It should not be an option to opt out. Being neutral on this topic does harm to learners.

Being neutral on the topic of power is wrong There is no such thing as being neutral. Yes, those are pretty strong words. I believe them. I will act on them.

I will put the rest of this “paper” below the fold, so it is not taking up tons of space. However, I encourage you to read on. I am going to try to become explicit in understanding what critical pedagogy is. I won’t apologize for that. It will be technical. And yet, I don’t see what I write below as optional practice in the classroom.

Critical Pedagogy / Theory: as I see it today, February 2016

Continue reading »

Feb 052016
 

Anne Schwartz wrote a wonderful post this morning about “Why I am not quitting teaching.” It is a extremely thought provoking post, and it made me think. She wrote many times “I am not,” “I do not,” etc, but it is easy to turn it all around and see the positive in every statement. Because, IT IS a very positive post, hiding behind many negative statements. She ends with,

Mostly, I am committed to never reading another fucking letter from a disgruntled teacher about their decision to quit.  If you feel like writing one of those let me know and I am happy to give you suggestions just where you can put it.

I made it a habit of mine to follow this advice daily.

Teachers inspire other teachers

Anne is one of those people.

But, I have been questioning myself the last 8 months because I kind of feel like I did quit. I am not in a classroom daily. I am not interacting with high school learners, and this has caused me to have a crisis of confidence. Am I a fraud, now? Did I sell out?

Anne’s post made me think deeply this morning. She uplifted me, and inspired me to think. And I realized something.

I did NOT sell out.

I moved into a different category of teaching. I didn’t walk away from my learners, and my colleagues. I didn’t walk away from the challenges of teaching, I embraced them more, and deeper than I did before.

I now have the added burden, challenge, and yes, job, of telling young adults how they should teach. Most importantly, I must communicate WHY they should teach.  And if I am not 100% clear to myself and the world that teaching is the 100% best use of my time, energy and effort, than I have sold out and am a fraud.

Screw that. I loved teaching. I LOVE teaching.

#WhyITeach

Because for 8 years I knew I can walk in the door of my building and know with absolute certainty that I worked with the absolute best people who cared about people. I may not have agreed with them every day. But no one could ever question their commitment to our learners.

Because it was about my learners. Every single day. It was about their success, not mine. It was about our learning, together. It was their bad days (because they had them) and my bad days (because I had them) but it was always how to get better and do better.

Because content is great, but people are better.

Because standards are important. You must know know where you are going, or you just wander around aimless.

Because my worst day teaching was still better than my best day in the private sector (10 years of that prior to teaching.)

Because I hated grading, hated grades, but knew they were a necessary annoyance to the process. So I focused on what was important, learning, and not the grades.

Because after 9 years in a classroom, I still can’t believe they paid me to talk math with people every day? And I had the privilege of talking addition with one learner one day, and calculus with another a different day, and stats yet again on a different day. And each day, each learner needed a different conversation.

Because each day was a new day. The blow up by a learner yesterday was yesterday. Every day was new. With a new beginning, a new morning, and a new opportunity to fix a misconception or a misunderstanding.

Have I “sold out” like I was accusing myself? No.

I have embraced it more. I have become a bigger cheerleader.

Now, I am in elementary, middle, and soon, high school classrooms. I am in college classrooms recruiting future teachers. I am telling them, honestly, why I teach.

Thank you Anne. You helped me resolve the internal, nagging voice that was telling me I was a fraud. (Damn Imposter Syndrome.) Screw that.

I teach.

I teach people.

I used to teach people math.

Now I teach people teaching.

That is so cool. Can you believe they pay me to do this?

Good morning. It is a new day. Let’s go teach someone.

Feb 022016
 

On Friday last week at the end of the Step 1 class we were talking about engagement, high fives, enthusiasm, and why we are teachers. The conversation started with these two questions:

  1.  Write about a lesson / teacher who you remember using a 5E model.
  2. Write about a lesson / teacher who you remember did NOT use a 5E model.

The conversation led to the idea that the teachers they remembered from #1 were teachers the learners in class remembered fondly, they remembered their classes with enthusiasm, and they remembered specific lessons from those classes. The teachers in category #2 were still good teachers (I did stress this) but the entire conversation was less enthusiastic. No lessons specifically were brought up, and it the words “favorite teacher” was never mentioned. As in, not even close.

And then I challenged the class. “What category of teacher do you want to be?” I let them think about it.

And then, I brought up the fact that I had high fived them the last two weeks. I asked why they thought I did that, and how did it make them feel. The conversation was epic. They realized how connected and interested just that one little thing made them.

At this point, after I explained my High 5 philosophy.

Then, as I do, I ask, “What other questions do you have?” That opened the door.

One question was, “Why do you call us learners?” Answer, because students study, and I don’t care how much you study. I care how much you learn, so I refuse to call you students. Also, if you have students in the room, then you also have …. [they said a teacher] … So, if I call you learners, that what am I? A learner, too. And I promise, I will learn as much from you this semester as you learn from me.

After I explained about my “learner” philosophy, someone in the class said, “You should make a mix tape.”

That statement stuck with me all weekend.

So, here is a mix of Waddell’s greatest hits. Now, before you say, “Well, anyone can be egotistical enough to write these,” know that I did not write these. I asked my learners from last year on Facebook to tell me what stuck with them. These are things they reported almost a year after being in my class. This is “My Mix Tape.”  My comments are in [].

——

“I don’t care how much you study, I care how much you learn.”

“If it was easy, it wouldn’t be worth your time.”

“You can’t memorize math, you have to learn it to understand.”

“Things don’t ‘cancel’ out.”  [did you know teachers use the word “cancel” to mean as many as four or five completely different things? This is a huge pet peeve of mine.]

But this is the whole quote of what was written: “Things don’t cancel out.” I know it was for math because things don’t disappear they become a 1 or 0 but it applies to life on how things don’t just disappear and cancel out. There are reasons, hurts, joys, etc that come. There is so much more to things than just “canceling” them.  [Seriously, can I cry now?]

“Amazeballs”

“You’re awesome remember that.”

“Use your Awesome brains.”

“You have all the knowledge, remember to use it.”

“Stop complicating things take a second look.”

“Once you know the basics to math, you know everything you need for any problem.”

——-

And, finally,
Learner #1: You were and are the most amazing teacher I have ever had
Learner #2: Can I second that?

 

—–

Okay, Now I have some tears. For realsies.

Jan 272016
 

I have not blogged much because I … well I haven’t been very reflective, more reactionary. I have been focused on building a program, recruitment, developing resources, and collaborating with my fellow Master Teacher, but I have not stopped and reflected at all on how this process has gone.

That must change. For me. So I can grow and get better at this new position (because I am in the second semester, and it really can’t be called new much longer.) That raises an interesting question. At what point is a “new” job no longer “new”? Hmm. I think about now, 8 months in, it is no longer new. It is me. And I need to stop and think; reflect; realize what I am doing poorly, what I am doing well, and start improving.

So what brought on this bit of soul searching? I read these two article in the same day:

Detroit teachers want you to see these disturbing images

Michigan Court: State Has No Obligation to Provide Quality Public Education

And, I am in a Critical Theory course for my PhD, which is challenging my perception of philosophy and what I am willing to accept as “normal.”

No way in hell will I ever accept that the content of these two article is “normal”. Teachers in Detroit are putting up with what? And Michigan judges have ruled (yes, in a very technical manner, but they ruled) that the State only needs to offer education, not a QUALITY education!

Oxygen Magnesium, are you freaking kidding me.

<<breath>> <<breath>>

Okay, this is not okay with me. I don’t live in Michigan, and this is not okay with me that this is going on in Michigan. It will never be okay in Nevada where I live. I will fight tooth and nail to make sure of that.

I am fighting to make sure of that. I will teach my new teachers about social justice, and Critical Theory. I will make sure they know to stand up for their learners and do what is right by them.

More blogging. More reflecting. It is important to me, and I have missed it.

Aug 312015
 

I have to be honest, I started, stopped, deleted, restarted, deleted and started this post again repeatedly over the last few weeks. Why? Well one reason is my computer died in the middle of a post, and it sat for a week while I was getting it repaired. Whatever. Lame excuse.

Another reason is that I was not sure what to say, or how I felt about the change from high school teacher to college instructor. I think I am still not sure, but I am wrapping my head around it more and feeling better about myself and my thinking on that topic. This post will be a bit rambling, and more than a little stream of consciousness, but bear with it.

So, here it goes; good and bad. I am going to just get it all out and see where it leads.

do not follow leave a trail

First, the bad: I felt very guilty about leaving my school. Seriously. The process of getting this position took all summer. The interview was a 7 hour long day in the middle of July, and it was a week after that before I knew if I got the job or not. Teachers reported back to school on the 5th of August. I was not able to give my school or my department much time to hire a new math teacher to replace me. I hate that. That I left my high school without giving them a long time to search and find a replacement makes me feel like I let the people who I had a strong attachment and bond with down.

The good: This new program at the University of Nevada, Reno is amazing. Seriously. Why is not every university in the US using this model of teacher development for math and science? I mean, really. We all recognize there is difficulty in getting math and science teachers. The UTeach model out of Austin, TX is a great model to fight the shortage. It is actually doing good recruitment and instruction to bring better math and science teachers to the classroom. Let me tell you the sales pitch (and it is a sales pitch that I have given to several freshmen classes.)

The Step 1 and Step 2 classes are free through a tuition rebate (after you successfully pass the classes, you get your money back.)

In these classes, you will observe twice, and teach three times in upper elementary (Step 1) and middle school (Step 2) classrooms.

At the end of the year, you will have two free credits, AND you will KNOW if you have an interest in teaching. If you don’t, because whatever, you walk away and you have two credits, no money spent, and you have lost nothing but a little time.

BUT, if you think that teaching may be something you are interested in, you finish the major you are in (right now Chemistry, Biology, Physics and Mathematics, but that will expand) AND you take the NevadaTeach program classes and you will graduate in 4 years with two degrees. Your science / math degree AND the coursework necessary for a teaching license.

Yes, free credits. Two degrees, two career paths, and no extra time or money spent to earn either one.

This program sells itself. We were expected to have 30 students in the program this semester. My partner Master Teacher and I recruited 45. We are 150% over the goal for enrollment. That is exciting, motivating and all around wonderful.

Then, we actually met our students.

OMG WOW.

On the first day of class (heck the ONLY day of class so far) we asked them to write why they took the Step 1 class. Here are a few, representative samples of why they enrolled:

I want a second choice if I can’t get into med school  (this came up several times.)

It seems like a fun program to be in, very excited about going into classrooms to teach an be like hands on.  (again, several of this type.)

I want to have my double major through this program and I think it will offer lots of opportunity in the future.  (wow, just wow.)

I want to explore teaching as an option.  (no fewer than 5 people said this.)

I’m taking step 1 because I want to have the best choice that allows me to have the best option to succeed in my future career.  (yes, this is the same as the last one, options, but notice the addition of choice. )

These are our students’ words. No editing. Just my comments in parenthesis. We have a motivated group of students who think teaching may be an interesting career. It is up to Megan and I to show them that it can be.

How do we do that?

One major element of our classroom and the program is that it centers around the 5E model of instruction. As we teach science or math lessons to our learners to teach to the ES or MS students, they are all 5E, inquiry based lessons. The math teachers who graduate from this program are going to have a strong basis for creating inquiry  based lessons for their classrooms. This is truly exciting. I am fully committing to dispatching an illusion of learning.

illusionoflecture1

What else is exciting is that this program did not exist last semester. I am part of the first year of creating the program from the ground up. If it fails, I will be a large part of why it fails. If it succeeds then I will be a part of why it succeeds (well not really, it can’t help but succeed.) But it is a risk to leave the safety of teaching, being department chair, teaching the courses I love, interacting with amazing learners and stop all of that for the complete uncertainty of a program that does not exist, in a completely different environment, and a radically different culture.

great achievements involve great risk

So, do I step up and leave everything I was comfortable with behind and bet it all on a new, untested, untried program to create and build new, more and better math and science teachers? Clearly the answer I chose was yes, but it was a tough decision. I miss the teachers I interacted with daily, but I know that I am doing something that will benefit more students in the future than I could just as a high school teacher.

As far as the massive culture shock, I have overcome it. Mostly. I have had a couple of “Am I on candid camera” moments. Being told “good job” for submitting $20,000 technology requests that were detailed and approved. Being told “ask for it, we don’t short change instruction, if you need it to teach, ask” by directors of the program. Coming from K-12 where we were starved for resources and now have the resources is odd.

Having to navigate the minefield of tenured professors walled gardens has been a shock. As a high school teacher, I just did things. I always could justify it because it was in the best interest of my learners, so there was never any blowback, just an “okay, that works, thank you.” Now, however, that is not always the case. And, what is in the best interest of my students is NOT the best interest of the departments students, the colleges’ students, or the University’s students. That is absolutely true. So having to think bigger picture and take a step back is new for me. Not hard. Just new.

The last thing that really is different for me is that I always sought out teachers to inspire me, to motivate me. As a high school teacher I lived by this quote daily.

Teachers inspire other teachers

My list was easy. Go on Twitter. Search for #MTBoS. Follow them. All of them. I have found so many teachers who pushed me to be better through their ideas, motivation, and inspiration that I never felt alone the last 4 years.

I am feeling alone now. I have a beautiful office. (seriously, it is the best office on campus, look at the view from my office window).

2015-07-27 17.56.52

 

I have a fellow Master Teacher, Megan, who is amazing. I have directors in my program who are supportive, helpful and all around great people. The faculty and staff here are supportive and helpful.

And yet, I feel alone. The college culture is different than K-12. There are no faculty plays. No “Friday happy hours.” No fabulous twitter chats of supportive higher ed professors. At this level it is about what you produce, not how you feel. K12 is different. I am working over that, around that, and through that, but it is true. I think this is the largest culture shock to deal with now. I can still drop into the Friday happy hour, but I am not part of that group. Will they still have me? And what am I producing now for my new position?

🙂

Yes, I just smiled. I realized what I have to make sure I produce.

Teachers inspire other teachers I need to be that teacher who uplifts, inspires, and drives others.

More so now than ever.

Jun 102015
 

Functions

I begin the class with a “what do you notice?” “what do you wonder?” session. This is probably the 5th or 6th day of class, and sets the stage for the entire rest of year. What do you notice? What do you wonder? I document all the noticings and wonderings, and then we discuss the mathematical questions.

Every year, the question of “I wonder how the 2 and the 1/2,” “I wonder how the 3 and the 1/3,” are related is asked. The best two questions that are always asked are, “Why are they all the same?” and “What changes when we change the exponent from 1 to 6?” I always say that I will answer every question by the end of the year; I will never lie to them and tell them something is impossible when it isn’t, but that some of their questions may need to be addressed in a future math class and not this math class. That honesty goes a long way.

I spend an entire period exploring the different functions with them, showing graphs on Desmos, asking for values to put in for a, h, and k. I ask questions like, “what do you predict the h will do?” and “Did your prediction come true?” The learners who are the typical aggressive type A learners hate it because they want the answers and want it now, but they will come around and start developing ideas on their own.

I start with lines for the (h,k) form because I think this form shows some reasons why to use the form, the benefits of using the (h,k) form over y=mx+b, as well as a simple function to cut our teeth on vocab.

I introduce this form first thing in the year as we get started. Fully Explaining & Understanding functions blank (double sided .docx file). I print off hundreds of this form, and we use it regularly. Some days I have the learners write the functions in their notebooks when I don’t have the forms, but I try to have a stack on hand always.

explaining This is what it looks like. There is A LOT of info asked for, and I start with lines so we can establish the understand of what the different elements are.

It always bugged me that we rarely talk about domain and range of lines. Why not? Why start introducing that idea with absolute value? Just because that is where it changes from all real numbers for both to only one, does not mean we shouldn’t introduce it earlier. Same thing with asymptotes and even/odd functions.

If I can get learners to identify the x and y intercepts on the line, and then connect those points with the standard form, so much the better.

Same thing with intercept and (h,k) form. Cut the teeth on a line, that is familiar and safe, so that as we move forward with quadratics, cubics, cube roots, etc, the learners can see the vocabulary does not change. What changes is the shape of the parent function.

I make sure every learner has one copy of this that is complete, pristine, written clearly and fully in their notes for every single function. When the learner puts them all side by side, they can see there is only one math, one set of ideas through the entire year. What changes is the amount of effort needed to get the intercepts for a cubic vs intercepts for a line? Why?

Another rich focus of questioning is “What makes the line unique?” “What makes the quadratic or cubic unique?” Some answers I have received are, “Only the quadratic always has a vertex form. The cubics can have that form, but usually not,” or “Every point on the line is a critical point, but we can’t always use every point for other graphs.”

Or can we? Hmmm. Leave it at that. Don’t tell them. Plant the seed and let it grow on its own.

This is a big picture post. Philosophy of teaching, approach to the topics, etc. No details yet. Just a pouring out of my thoughts on how I start. I will go more in depth. Notice that there is not enough room to work on the page. Only the results go there. The work is separate.

 

Jun 082015
 

My learners have been using Plot.ly for a week, and have asked me a ton of questions on how to do certain things with their data. I wanted to add details to my last post on Plot.ly v. JMP and tell you the decision I made regarding the issue. All of the questions I have below are actual questions / issues  my learners ran into using Plot.ly.

Issue 1. How to add % totals to the columns of data in a graph?

One group of learners had a beautiful graph made in Plot.ly. It was nice, communicated well, but had lots of information in it. They wanted to put the % of each column in the graph to make it more informative.

In other words, they had this ……….and wanted this. (the reason for the arrow in a sec)

graph1 graph2

Yes, these are JMP graphs. Why? Because after an hour of looking, I could not find a way to have Plot.ly do it. Their help is silent on this issue, and I looked through a whole bunch of graphs shared on their website and found not a single one to do that.

As far as JMP, it took two clicks. I can’t show the menu because it is a drop down and as I tried to screen cap, it went away. You click the red triangle I pointed to, hover over to “Histogram Options,” and click on “Show percents.” If you want to “Show counts,” you can do that too. One or both! Two clicks. This was incredibly simple to do in JMP, incredibly difficult in Plot.ly.

Issue 2: Chi-Square test

I already dealt with the fact that Plot.ly calls graphs that use categorical information histograms in my last post. This has caused so. much. confusion.

But now my learners are trying to do the statistics for their data and see if there are significant differences in their samples. They are trying to DO statistical inferences. If their data is quantitative, they can do a t-test easily. Well, they can do a two sample t-test easily. They cannot do a one sample t-test or a matched pair t-test. They cannot do a z-test in Plot.ly, and as it turns out, you cannot do a Chi-Square test in Plot.ly unless you already have the summary counts.

Really? I can do the “histogram” to get the counts, but I cannot import those counts into the table to do the Chi-square? It won’t count the instances of words to count them for the test?

For example, if the learners data looks like this:

data1  Plot.ly will do a histogram for it and tell me what percent or what counts there are for Gender and AP/Honors.

If I want a Chi-Square test for these two columns, the only way I could make it work was to look at the graph of counts, write down the information into a two-way table, and enter the counts as a matrix in the graphing calculator.

To do the same thing in JMP, we do the following steps:

1.  Go to Analyze, Fit y by x JMP1

 

2. Click on OK. That’s it. The output contains the following:

JMP2  A mosaic plot of the graph which is nothing more than a stacked bar chart, except the width of each column is proportional to the total number of things in the column.

Next, we get the contingency table. If I click the red triangle, I can choose other values to include or exclude from the table.

Finally, the Chi-Square test p-value.

That was around 6 clicks, instead of making the graph, counting from the graph and writing a table, and then inputting the table to the calculator.

Issue 3: separating data by a response

The group who was doing the AP/Honors and work in Issue 2 had another problem. They asked for GPA and the number of hours you worked. But they needed the mean GPA of only those in AP/Honors and those not in AP/Honors, as well as the number of hours worked.

Plot.ly will give us the total 1 variable stats for the column of hours worked, but it will not give it to us in two groups of Y/N based on type of classes taken. It will not do it.

Enter JMP. 6 clicks. Analyze, Distribution, put the variable where you want them, OK.

JMP5

That’s it. You get a 1 variable stats for those who are in AP/Honors, and a separate 1 variable stats for those not in AP/Honors. Doing a two sample t-test is simple and easy once this information is obtained. This is not information Plot.ly can give us.

Issue 4: Linear Regression t-test

Last issue, and then I will stop. I have several learners doing quantitative projects that lend themselves to linear regressions and linear regression t-tests.

Plot.ly makes beautiful scatterplots. You can adjust the axis, overlay the regression line, insert the equation into the graph, etc. They are pretty.

But, if you want a residual plot. No go. If you want to reinforce the statistics of y=a + bx. No go.

This is what it looks like in Plot.ly.

plotly1 You have y=mx + b from algebra, you cannot do residuals, and you CANNOT do a linreg t-test.

In JMP, it looks like this:

JMP4 5 clicks, Analyze, Fit Y by X, put the variables in the correct spots, and hit OK. Notice this is the exact same dialogue box you use for categorical data. JMP uses the same path for different types of data, but tells you in the bottom left corner HOW it will act on your data.

You get output that looks like this:

JMP3 If you want the residual plot, hit the red triangle next to “Linear Fit” and show residual plot. That easy.

Bottom line

Although I fully understand that every single complaint I have had with Plot.ly can be solved by learning the programming language and learning to program the software, I don’t think I can ask high school learners, in the last 4 weeks of class, to learn it so they can do a project on statistics. Honestly, I don’t want to take the time to learn the programming language of Plot.ly so that I can do it for them, either.

Plot.ly makes BEAUTIFUL graphs. It is a powerful platform to show connections between quantitative data sets. But, it does a so-so to bad job on statistics.

JMP makes graphs that may not be beautiful, but the statistics is primary to the operation of the program and makes doing the statistics easy. I think without some major changes to Plot.ly to work towards the statistics side instead of the data representation side I will go back to using JMP next year.

It was just too difficult to teach the way Plot.ly handles or mishandles the stats.

 

Jan 172015
 

Nothing annoys me more in teaching math than a bunch of rules to memorize, and rational function come with their own complete set of rules to memorize. I really find that annoying, and I have been on a personal quest to make sense of algebra through a combined set of understandings that will bring comprehension, not rule following.

I have found that in large part through the (h,k) form of the algebraic functions (and here too). Not just a little, but the (h,k) form now drives my entire instruction to the point where my learners are asking me first “how do we undo this” instead of “what chapter is this” as we are learning the math.

So, rational functions. How do the “rules” of horizontal asymptotes fit for rational functions. I really struggled with this the first year I was working on the translations and (h,k) ideas, but this year it all fell into place.

Lets take two functions, f(x) and g(x) where the highest degree is m for the numerator and n for the denominator (just keeping things in alphabetical order).

The rules that everyone knows and hates:

If m=n, then horizontal asymptote is: y=a/b where a and b are the leading coefficients of the numerator and denominator.
If m>n, then there is no H asymptote [or some books say if m=n+1 then there is a slant asymptote]
if m<n, then H asymptote is: y=0.

Okay, I hate these. I really wanted to understand why, and I fully understood when I explored how to get any rational function into the (h,k) form. How do you do that, you ask? Simple. You do the long division and rewrite the equation in the new form.

First off, though, we need some functions to explore. I have a Desmos file with 1600 different possible rational functions:
 Seriously, 1600 possible functions. 40 for numerator and same 40 for denominator.

I tried typing it all out, but failed, so I wrote it out and took a picture:

2015-01-17 16.05.06

What we see is that the ‘k’ value is always the horizontal asymptote. What we also see, is that there is ALWAYS an asymptote when m>n, and sometimes it is a linear slant. It also, can be a quadratic slant, or cubic slant. What is important is that the horizontal asymptote is a way to discuss the END BEHAVIOR of the curve. If we have a slant asymptote, what is happening is the original function is approaching the value of another function instead of a constant.

Rock my world.

So, 2x^4 +3x^3-2x^2 + 5 divided by 2x^2+4x-2 gives us a ‘k’ of x^2 -.5x +1. The “slant” asymptote is a quadratic function.

2015-01-17 17.14.21Here is the math:
 and the Desmos file.

What is amazing here is the long division and putting the function into (h,k) form means you do not have to remember ANY rules with rational functions. It also means there is a reason to teach long division of functions as well.

If our goal is to create a unified, sense-making structure in algebra, this is how it is done.

Let me know if I have made a mistake somewhere or there are flaws in my thinking. This is one piece of the larger structure I am seeing with this approach to algebra, and I really want to push the envelop and limits of of the method.
At this point, what I see is that the “rules” of horizontal asymptotes are nothing more than tricks. The math is the long division and rewriting the function into the (h,k) form to show the translations, and reflection.

In addition, if you look at the functions I used in the explanation above (the first picture I used), you will see that only when the function is put in (h,k) for does the reason for the reflection show up. If the function is left in standard form, the reflection is hidden.

Nix the Tricks! This is the reason.