# Would you Rather: Hypothesis testing

I am teaching Type I and Type II Errors Monday and wanted to get students thinking through the problem of balancing the probability of the two errors. I want them to understand the tension and realize they will not be able to eliminate either possibility. I plan to do this before actually introducing the names for the types of errors. I just wrote up these two “Would you rather?” scenarios. I would love some feedback or some other suggestions to use as follow up.

For the two scenarios below, decide which error would be worse. Clearly state your answer and your reasoning.

Scenario 1

As a doctor, you see a large national study that 35.9% of Americans are now considered obese ($H_0:p=.359$). Alternatively, you think it may be possible that your patients are below the national average ($H_a:p<.359$). If you conduct a hypothesis test on a sample of your patients, which would be worse?

a.Rejecting the null hypothesis and stating that less than 35.9% of your patients are obese when in fact your patients are in line with the national average.
-or-
b. Failing to reject the null hypothesis that your patients match the national average, when in fact less than 35.9% of your patients are obese?

Scenario 2

You are a researcher. Patients using the current treatment for lung cancer go into remission 55% of the time ($H_0:p=.55$). You believe that you have found an improved treatment ($H_a:p>.55$). Which would be worse?

a. Rejecting the null hypothesis and stating that your treatment is better when in fact it is not.
-or-
b. Failing to reject the null hypothesis that your treatment is equal to the current treatment, when in fact it is better.

# Guerrilla Students

My coach (the person who observes me frequently and gives me feedback) has complimented my students’ “scrappiness.” I love this categorization of them as it reveals their persistence in the midst of a system that I often think has not served them well. The other thing that I appreciate about the honesty of this categorization though is my students’ guerrilla attitude toward schooling. I think of street fighters who are ill-equipped, and so adapt various strategies to survive. I think many of my students have evolved this approach to their schooling.

Unfortunately, I believe many of their survival strategies limit their growth potential to “just surviving” rather than thriving. A colleague and I want to come up with a way to give our students the necessary tools to continue growing, not just get by.

So here’s what I want to know: What do you consider the most critical skills for students to have to succeed well in a challenging problem solving environment?

I think I want to emphasize as my big three

1. Collaborating
2. Valuing the process over the product
3. Communicating clearly and concisely

Anyone have any experience teaching students these things? Do you have ways you build practice in to your classroom?

# Differentiation pt. 1

Pardon me while I continue to idolize Sam Shah. He’s inspired me quite a few times recently.

A couple days back, Mr. Shah posted a project he was planning to use for his students to foster extracurricular exploration of math. Quick summary: There are TONS of cool math sites and blogs and newsletters on the internet which highlight progressive, artistic, fun, puzzling math, and he put together a project for students to choose several things to investigate.

As I read it, I LOVED it. I replied to him on Twitter and soon realized that another favorite math teacher blogger had been his inspiration. SO MUCH COLLABORATION!

As it is my first year teaching all the classes I am teaching, I am trying to hold off on making a huge investment in forming projects at least until I have a chance to get through a first year and make it to the summer to have a chance to reflect. I began to think of ways I could still use these ideas in my class.

Enter tension 1: In a recent observation, my observer challenged me to go all in with differentiation. Its always something I struggle with, as it tends to make me think I have to plan separate lessons for each level student in my class. I have no more time for that. However she introduced the idea of just offering choices in how students show mastery. No doubt its a lot easier to give a problem set and get all students to work through it, but she suggested a choice board. This one is from an art class, but the point is to create options that engage students from the different learning styles and preferences.

Enter tension 2: I usually teach to the majority of low level learners I have. I have not been happy with how I am (not) challenging my high level learners. I feel like the common advice was to turn them into peer tutors when they finished their work. I know they can benefit from that. I am learning a ton about statistics by teaching it for the first time this year. I’ve always felt a bit uneasy about that idea. I want them to be excited to finish other work early so they can continue to challenge themselves with a new and exciting topic.

Tension 1 + Tension 2 = My Current Idea

I am trying on using most of Mr. Shah’s mini-exploration activities as an opportunity for choice and for my students to continue to push themselves. So far, I’ve had a only few students I’ve asked to participate. I ask them to go to our class website and read the directions and pick an activity to complete. See the work in progress. I’m still working to break it down into clearer chunks. Vi Hart’s doodling has been pretty popular so far, as well as a couple of the games on Math Munch. Still looking forward to see how it works out as time goes on and as the routine continues to settle in.

# Introducing basic trigonometric ratios

I take no credit for the idea, but I am absolutely sharing the results. Thanks to Mr. Shah, a rock-star teacher blogger, and his Virtual Filing Cabinet, I had a wealth of ideas at my disposal when planning a trigonometry unit for the second time this year.

Last year, I started with special right triangles. We then for some reason detoured into arc/sector measurements before returning to me telling them the patterns that these “special triangles” could make in the unit circle. YAWN!  I asked my students to memorize the unit circle, which I am now not convinced that it is that important anyway (Although Ms. Gruen’s blog gave me a great memory tool to help students remember unit circle values). Then we used a great activity from TI Education to build the sine and cosine waves. I stand behind this one, although I already know this year it will make SO MUCH MORE SENSE to them.

The reason I am optimistic about the TI activity is now all rooted in the new approach I took this year. Courtesy of Mr. Lark, my students were able to find their own patterns and shortcuts. Mr. Lark’s activity takes a lot of the heady, abstract nature of trig and returns to a bunch of simple calculations. Students measure angles, draw radii, and write the x- and y-coordinates of the points at which those radii end. By the time they get really tired of doing this over and over again, they can start to see patterns that they can latch on to. I introduced my version of the activity (see below) with a very explicit exhortation to use the patterns when you find them. After a couple days, we discussed and students suggested patterns that could be later tied into:

• reference angles
• why sin, cos, and tan are positive and negative in different quadrants
• symmetry between the various quadrants

I’m drooling at this point. My students are thinking like mathematicians and all while working on an activity that really only required knowledge of how to use a protractor and how to read the coordinates of a lot of points. I then make the big reveal: “I know you’ve heard how miserable trigonometry is, but YOU’VE BEEN DOING IT ALL ALONG!” The stares I get back say, “Who is this crazy person?” but then I explain how the x-coordinate they were finding over and over is cosine, and the y-coordinate is sine, and the ratio of y/x is tangent (which I somehow didn’t realize until this year is the slope of the radii in the unit circle formed by that angle!).

We moved forward and derived the SOH CAH TOA general trig ratios by showing how any right triangle was proportional to a unit circle triangle. I am really impressed with how well my students have latched onto the sin, cos, and tan since then.

It’s also pretty cool when you hear “I didn’t know SOH CAH TOA actually meant anything! I thought it was just something people said…”

Now that I am getting to introducing the sine and cosine waves this week, I’m looking forward to lots of A-ha! moments from students remembering that cosine is the x-coordinate and sine is the y-coordinate on the unit circle.

# Failing Forward

A colleague of mine (@JoshShelleyUNC) shared this NPR story with me a while back on how students see struggle differently in Eastern and Western cultures, and I have tried to use it before in my classroom to send a message to my students: struggling is not an indicator that you are dumb. I loved the story, and I think it made a brief impact on the students’ thinking, but it wasn’t very sticky. I couldn’t refer back to it easily, and it just wasn’t captivating to them. I listen to NPR a lot, but it didn’t really move the needle with students who think life should be about their entertainment.

I just finished reading an essay by Keith Robinson, a teacher from Newark, NJ who was recently awarded a Fishman Prize from the New Teacher Project, which rewards “superlative classroom practice.” He wrote an essay titled “Gettin’ Messi: How Mistakes Make Mathematicians.” (find it here, starting on page 21).

He discusses how he introduces students to Lionel Messi, the world’s greatest soccer player, and after wowing them with his skills, he breaks down their fixed intelligence mindset by showing them video of how hard he practices. He shows the breathtakingly skilled player taking free kick after free kick and missing many of them. The essay goes on to discuss how from that day forward, Messi is the class icon for persistence and a growth mindset. He discusses how he creates a place where mistakes are valued at a higher level than correct answers because of the potential growth involved. For the full perspective, read it yourself. I love his idea for a couple reasons: 1) I’m a Messi fan (have a jersey and all), 2) its STICKY!

Then I reflect on my classroom. What have I done to engage my students in their mistakes in a positive way? Well, I have a sign on my wall that says Fail Forward. I have said a few times this year that students’ mistakes are more important to me than the right answers. I had a couple assignments earlier in the year where I had them look at intentionally incorrect work in order to find the mistake and make recommendations on how to improve.

I have done a little bit, but I know I’m still a long way away when I do a TI Nspire quick poll with my class and have some students asking “Who put _____?!” They’re interested, but it’s clear they have no interest in helping that person out. In a culture where students bond by making fun of each other, I see an uphill battle.

This element of class culture is especially critical in my classroom. I teach mostly junior and senior math classes where I have a HUGE range of ability levels. My pre-calculus classes are all labeled honors, but since there are no “regular” sections, I have students who have been passed through the algebras and geometry without much comprehension and I have other students who are legitimately deserving of the honors title. In that classroom, the practice of students comparing themselves to one another can be devastating if students see their ability something they have no control over. Students giving up becomes much more common, and there is no reason to push themselves.

Since I believe the goal of the education is creating thinkers, it is critical to have these students be self-motivated. It is impossible if they see their mistakes as defining their ability. If they truly believe that their efforts can make them better, they are empowered to face head on the many obstacles they face daily. The challenge now is to show them a model of humility and working to improve on mistakes by addressing this ASAP once we get back from fall break.

# Rich Problem Solving – Experimental Design

I’m currently in the middle of Fall Break. As each moment goes by outside of school I regain a bit of my prior sanity, and am able to reconnect to goals and heart of my teaching philosophy. Conveniently, the week of fall break also coincides with the first challenge of the Explore the MTBoS Mission 1: Explore the power of the blog.

I found out about this challenge as school was starting and planned to be blogging all the way until it started and use it as a jumpstart to keep me going. Now, instead, its the jumper cables that hopefully will push me to keep growing during what has turned out to be a REALLY busy and stressful year. Turns out three new preps take up all my time.

I want to respond to the first challenge in brainstorm mode, as I feel that I have not done a great job so far this year (in any of my three preps) of teaching through rich problems. However, in AP Statistics in particular, I want to continue to replace the math in their heads (a loose collection of skills each with an associated step by step process) with struggle and critique and logic.

One overarching topic of the AP Statistics class is experimental design. Students are to engage in the art of creating surveys, studies, etc. in order to minimize bias and examine conjectures statistically. As the majority of my students have little exposure to reading about research, doing or taking surveys, I am interested to throw them in the deep end and see what they think is reasonable for experimental design.

I envision setting up a class near the beginning of the experimental design section as a role-playing scenario, where students are asked to take on the role of researchers trying to answer some big question about their community. In the beginning, I would give them little guidance, just a promise that throughout our unit we will continue to improve this research plan and build up to the point where we may actually be able to carry out a survey or experiment at the end of the unit that gives reliable data. As we discuss ideas such as bias, sampling, and blocking,  I would like to allow students time to synthesize the new ideas by making successive revisions to their research plans.

What I envision struggling with here is scaffolding the initial brainstorming activity for my students. The urban education system they have grown up in has made it very difficult for them to speculate, apply, or create new knowledge without having very explicit modeling first. I want to push them to move into a new topic even though they have little background knowledge and be willing to put something on paper even though they know it will not be good.

I’m thinking I will have them brainstorm based on a series of basic questions about designing an experiment. For example, “What data are we trying to obtain? How will we obtain that data from our participants?” and so on. Maybe I will have them read articles about research for a few days leading into this topic in order to answer questions about the researchers purpose and methods.

Any better ideas on how to scaffold them into this or more questions I could have them work through? Any scientists out there have a sample of the things that you would do to plan a study before you do it?

# The Only Spring of Knowledge

I teach in an urban district, in a poor neighborhood where >90% of our students are on free or reduced lunch. The schools in our neighborhood have been making strides, but have regularly been on the state’s naughty list with regards to success. There are already enough discussions of why it got this way and how to fix it and who to blame on the internet, and I don’t have the time to waste on that anyway. I choose to focus on my classroom, and what I can do (or can’t do).

I’ve spent a good deal of time in the last couple months reading and taking ideas from some prominent math teacher bloggers from around the country who are true innovators in terms of content and rigor of their courses. One of the biggest revelations of getting caught up in math teacher blogging world is realizing how limited most of my students truly are. I know its oversimplifying it to say it this way, but most of my students are functionally illiterate! They could read through a passage and say most of the words correctly, but if there is a new word (particularly a math term) they struggle to sound it out. Then if I ask another student to rephrase or summarize, I first get word for word quotes. If I keep pushing, maybe that one particular kid can save everyone else from their dumbfounded silence.

They didn’t teach me this in my undergrad, but illiteracy is equally as disabling in math as it is in English, history, or any other course that typically carry the torch for reading and discussion.

When students choose not to read because they don’t understand it anyway, they become primarily auditory and visual learners. They want to hear everything or see pictures. I don’t mind catering to those needs and am doing my best. However, it leaves me as the only spring of knowledge in the room.

I have tried to adapt group work packets from some of my favorite math bloggers like this one (which I stole almost completely from Kate Nowak), only to have my classroom devolve into social hour because the students are busy waiting for me to come around and explain the questions to them and walk them through step by step, even when I have written out hints beside the questions. Many never learned to struggle through a problem and evaluate their answers based on other resources.I think this is one of the things that makes teaching so exhausting for me. I have to do everything for many of the students.

Part of this is venting, but also a call for ideas. How would you react in this situation? I know there is no magic bullet, but I want my students to realize the value of becoming more self-directed and responsible learners. How would you communicate that?