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.

Counting, adding, multiplying, and powers are just repeats of each other

This past winter, I somehow was led to an interesting insight that the basic functions in math are simply repetitions of each other.  Addition is just repeated counting, multiplication is simply repeated addition, and powers are just repeated multiplication.  The mental picture breaks down with weird stuff like negatives and fractions, but it can still form a nice intuition basis for WHY we wanted to invent these functions as an earth family and why they continue to be useful to us today.

Since the concept is a little abstract, and I wasn't sure if I fully believed it myself, I wrote a simple program in Python that has four functions that call each other.  For those of you new to reading Python code, here are the key things to know:

• Every function starts with a name and inputs.  To declare that it is a function, "def" is used to start the line.  Then the function name follows.  Finally, in the parentheses, are the input(s) passed into the function.
• At the end of a function that computes something and wants to send it back, you need to create a "return" statement.  This simply sends back the answer to the place that called it.
• On each line with an equal sign, the statement to the right is executed first.  Then it is saved in the variable on the left of the equal sign for later use.
• The statement "for i in range(__)" is a Python way of saying "loop __ times".  The things that are looped are indented one extra tab from the "for..." line.
• Finally, the "#" sign is not a hashtag!  It starts a comment on that line.  This means that everything after the "#" is NOT code, but an explanation of what is going on for the reader.  When writing the code on my computer, the text editor changes its color (as I have also done here) to help you notice that difference.

def count(initial):  # the count function takes an initial value
return initial + 1  # it adds one and return the result


def add(val_1, val_2):  # the add function takes any two positive integers
answer = val_1  # start counting from the first value
for i in range(val_2):  # the number of times to count forward in the loop
answer = count(answer)  # count once on the old answer
return answer  # all done!


def multiply(val_1, val_2):  # the multiply function takes any two positive integers
answer = 0  # start the repeating adding from 0
for i in range(val_2):  # the number of times to add in the loop
answer = add(answer, val_1)  # add once on the old answer
return answer  # all done!


def power(base, exponent):  # the power function take base and exponent
answer = 1  # start the repeat multiplying from 1
for i in range(exponent):  # the number of times to multiply in the loop 
answer = multiply(answer, base)  # multiply once on the old answer
return answer  # all done!


When I call "add(7,5)", it returns "12".  This required the program to call "count" 5 times.  When I call "multiply(3,6)", it returns "18".  This required the program to call "add" 6 times, which in turn resulted in 18 calls to "count".  Finally, when I call "power(4,3)", I get "64".  This required 3 calls to "multiply", which made 12 calls to "add", which made 84 calls to "count".  The point is that the higher order functions all rely on the ones below them and just add one more layer of repetition.

If you're reading this, I hope this sparks some new ideas for you!  I would also love to hear if you have ideas on extending this.  It would be so cool to fully build up algebra in a computational world!

Automathography

Justin (@j_lanier) runs a smOOC (small open online course, as opposed to the massive ones) called Math is Personal.  The idea, as I understand it, is to help us reflect on our own relationship with math throughout our lives and to build on it in a way that makes it easier to connect with students in our classrooms.  Our first assignment is our math autobiography, or automathography.  Though there are many ways to tell a story, I didn't know the big themes until I finished writing, so it came out as a timeline.  I'm not going to pretend this was written for a public audience to quickly skim and learn something new, but if you have some time, I would love to hear how it connects to your own story with math or where you would push me to go deeper in my own reflection.

As far back as first grade, math was my favorite subject.  I was fast at it -- give me 5 minutes to do as many addition facts as I could on a sheet of 100 and I would hand it back to you done in 2 minutes.  We also did a math activity each day called Number DAP, essentially a solo activity to arrange manipulatives in a way that demonstrated understanding of a concept.  I would always finish one task quickly, but despite often finishing a second task during our work time, my teacher didn't make it back around to me since she was still checking off classmates.  This drove me crazy, since to me, math was a race, and I was super competitive.  At the end of first grade, my teacher moved me into the 2nd grade math workbook where I could work in the back of the room.  Two other students joined me shortly after, and by 2nd grade we were permanently moved a grade ahead in math.

In 2nd and 3rd grade, I worked in the back of the room during math time with my 3 other math buddies -- Sonja, Alex, and Kirsten.  Kirsten is now my wife, but since I didn't like girls at all in elementary school, there was no drama in the back of the room.  That said, we did have interesting dynamics.  Whenever I finished something before the others, they made me share my answers under threat of being poked with their pencils.  We never really got a math lesson -- we just got a brief concept presented to us and used the examples in the workbook and one another to figure it out.  It was nice to not have to slow down.

Moving up to 4th through 6th grade, we did our math in the classroom with older students.  This adjustment was fairly easy.  One of my favorite parts was having to walk to the middle school across the park each day with our growing group of advanced math students to attend class.  I was a shy kid, so socially this was actually pretty helpful for me to have a group with boys and girls that I talked to every day.  It was extra awesome when 6th grade was moved back to the elementary building so we had to walk over two years in a row!

My first deep struggle in math came from multiplying fractions.  Multiplying is a function used to make things bigger, but my teacher was up in front telling us lies about how 1/2 times 1/3 was 1/6, a number SMALLER than either starting value.  I just didn't get it and kept raising my hand to argue.  I felt bad since nobody else was raising their hand (which to me meant they all understood it), but it was so shaking to my entire understanding of math that I had to be heard.  I think in the end the word "of" turned the corner for me, such as taking half of 1/3 would make it smaller, but it still took a patient teacher and a lot of examples to come around.

Middle school math was not too bad either.  Most of the advanced math students took Algebra (not sure if it was a true Algebra 1 or more of a 8th grade type of math), and then in 8th grade learned Geometry.  That was my toughest class to date with all of the theorems and proofs.  Logic didn't feel like math to me, but was instead a challenge in organization and argument.  I still liked it, and my teacher was fantastic (though he didn't put up with any shenanigans from our crazy cohort).

My high school had a traditional schedule, but some of the math classes were offered in block.  This would give me the opportunity to go through math even faster after being slowed to a single rate for most of elementary and middle school.  Algebra 2 was a pretty easy start to high school -- the class moved pretty slow since most of my class was juniors who didn't understand math well.  Sophomore year trig was much more interesting and pushed me to think more, but the block format of the class gave me enough work time to avoid homework most nights.  Later that year, I had a brief stats class filled with a lot of graph sketching (which I found incredibly boring since I didn't like making neat, ruler-guided lines for everything).  Pre-calc seemed like a painfully detailed version of algebra, especially things like limit definitions of derivatives, and homework took a while.

Junior year rolled around with Calculus.  I hardly knew what it was until I was in the class, but I knew I wanted to get to it.  I didn't even know what came after it -- it was the highest math class offered in the building.  It was also the first time I had to work my butt off, every single day, to do well in math.  I'm not sure what about it was hard -- I didn't mind the algebra, I had a TI-89 (I was a bit of a calculator nerd at this stage of my life), and the visual part didn't scare me -- but every day was a struggle.  One side note about how I wrote out math problems in the past -- I showed as little work as possible and wrote as tiny as I could.  I prided myself in fitting 5+ complete assignments on a single sheet of paper.  Since math was about getting answers, that was all I needed to write.  This made a handful of teachers mad over the years, but most eventually caved since I could demonstrate that I knew how to do it.  Getting back to Calc class, this wasn't going to work.  However, I still couldn't bring myself to use a full sheet of paper or more for an assignment, so I turned to whiteboards.  Everyday I just stood up during work time and did my problems on the board.  Quickly I was joined by a few others who liked standing and writing big while doing math.  As a result of writing large and doing so next to peers, this was when I started to do a lot of peer teaching and learning.  I loved to explain my thinking to somebody who was struggling and I liked working through problems as a small group in class.  Before Calc, this would have slowed me down, but now it was actually saving me time and helping me understand.  At home, I spent 1-2 hours on my Calc homework most nights, learning almost everything by example as I worked through the solutions on a website that had a step by step guide to every problem in our textbook.  Some people criticize this method, but it worked really well for me to develop a solid understanding.  I still learn best by seeing lots of examples and drawing my own generalizations and conclusions.  Despite how much work it was, I must have enjoyed it since I always did it first (probably why I ended up using Sparknotes for a lot of my English novels instead of reading them).

In my senior year, I signed up to take Differential Equations at the liberal arts college in town.  This was when I stopped liking math.  Not coincidentally, it is the first place where I couldn't use effort to overcome a lack of understanding.  There was a hard-to-follow lecture with no answer key and homework problems that assumed prior knowledge of physics (which I had not yet taken).  Eventually, I managed to wedge my way into a homework study group on campus at night.  I don't know how my parents felt about me being out past midnight on a school night on a college campus, but that was my only path to survival.  I contributed almost nothing to the group but was so thankful they let the little high school kid tag along.

A couple things that were going on during high school that shaped my experience were my deteriorating vision and my participation (and evangelization) of math team.  When I was 21, I was diagnosed with a rare eye condition called keratoconus.  Before then, I just had my mom and my friends all telling me I was blind, but when I went in to get glasses, they never improved my vision (I see okay now thanks to a hard contact layered on a soft one in each eye).  As a result, I was in denial about my eye problems yet could not see the board in class without squinting, so I never watched the teacher's examples that closely and often developed my own similar ways of solving problems.  As for math team, I joined my freshman year and loved an event that was challenging, competitive, team-based, and something I was good at.  By the time I was a junior, we had new advisors that worked hard to get lots of students to attend the monthly meets (mainly through extra credit), so it was extra exciting to be on the "varsity" group whose score counted for our school point total (I treated it like a real sport, to my friends' amusement).  I was so enthusiastic about it, the advisors asked me and another energetic senior to walk around to the math classes to promote how awesome math team was.  In the meet hosted at our school, we managed to get almost 70 students to come, so we were all pretty excited.  Despite the many interesting, non-traditional problems I was exposed to through my math team participation, it was the competition and the team aspects that motivated me the most.

College was a complete pivot in a lot of things for me, including my involvement with math.  I went to Olin College of Engineering, a new design and project-based school of ~350 students near Boston.  Math class was fully integrated freshman year with programming and physics, and all students were in the same course together.  We were asked to do a lot of open-ended tasks, mostly around modeling and simulating physical scenarios.  My goal of getting into a good college and racing to do as much math as possible were no longer relevant, so I lost my drive to push for higher and higher levels of math.  I took a greater interest in circuits and programming, both of which used some math, but the math was simple and used in a completely different way.  I actually avoided pure math as much as possible.  The low point was taking Diff Eq (the same one I took in high school) with a visiting professor -- I once again had almost no idea what was going on and once again got a C.  My favorites were Discrete Math with a professor who worked for the NSA for a while and Statistics with one of my programming professors (read his free and amazing textbook).

College was also where I developed my obsession with education.  In high school, I always enjoyed tutoring others and once tried to write a Calculus textbook that explained things in a much simpler way than our class textbook (that project didn't get too far), but I never truly considered teaching as something I would actually do as a career.  Near the end of my freshman year of college, however, I suddenly became on fire not for what I was learning, but how I was learning.  I loved the hands-on, fully integrated way of learning that I never got in high school.  I wanted to design schools that would teach this way.  I took one class my sophomore year, Improving Schools, that convinced me that there were already a few awesome models out there that could be built upon.  I was so into education-related things that I was running out of time for my engineering coursework, pushing me to take a year off of college to dedicate myself fully to education.  I spent about 15 hours / week preparing for and teaching a Saturday STEM class for Black and Hispanic high school students west of Boston, but most of my time was spent working on a ed-tech startup with a group of five other friends who also took a year off.  In the end, the business didn't pan out, but I was determined to stay in education to try to figure out the real problems and hopefully some solutions.

Three weeks after graduation, I got married, so I knew I would be moving to Minnesota where my wife was finishing a physician assistant's program.  I found a one-year teacher certification program through Winona Stats University in Rochester where I could take a summer of classes but spend the majority of the year learning by doing in the classroom.  I wanted to be a science teacher, probably physics or chemistry, but my college coursework didn't show the right prerequisites for the program.  Only by being a math teacher could I get certified and in the classroom quickly, and since that was my main goal, I went for it.  I figured I would spend three years teaching and move into administration as soon as possible so I could get closer to my bigger goal at the time, starting a new school.  I'm so thankful that I decided to actually put myself in a situation where I could be in the classroom as a teacher every day for the past three years, because now, I appreciate how much more complicated things really are.  Before I would even consider trying to start a school, I want to see what kind of innovation, especially in math education, can be done in our traditional system.  Though biased from working in a forward-thinking district, I'm finding that many of the limitations are not from "the system", but from our lack of imagination on how things could be.  That's the problem I've been struggling with for three years and counting.  One big piece of the solution is a much deeper, richer, more nuanced understanding of math and how we can think about it and use it.  That's where I hope my journey as a math learner takes me next.

Intro Stats Unit Idea

I have lots of projects in Stats, but the start of the course is always a bit flat without an engaging, overarching project that connects surveying, sampling methods, summary statistics, graphs, and confidence intervals.  All of these topics use one quantitative or one categorical variable and encompass the full process of collecting, analyzing, and presenting data (exactly what I look for in a good project).  It is a set of topics that shows up in the high school required standards (Minnesota and CCSS) and is foundational in AP Stats.

Last year, I carved a lot of time out of the class for students to define their own projects (around 40% of the course).  For a variety of reasons, I plan to cut back on that open time, but I will build in some of the better ideas into the core curriculum.  One idea that students had a lot of fun with was a "would you rather" survey of seniors at school that they turned into a visually appealing infographic (click to enlarge):

As a kick-off activity on the first days of class, I like to do some kind of simple survey using Google Forms to get students used to the idea that questions are variables (the response set), these variables come in different forms (most notably quantitative and categorical, but also ordinal and qualitative), and people are individuals (which are the rows in a spreadsheet).  I think this would still make a good opening activity, but now we would start a questioning each part of the rushed process after we finish:

• Who did our data actually represent?  If we did a convenience sample of most of our class, it hardly even represents the class well, let alone the school or all US high school students.  What if you wanted to talk to the same number of people and yet say that your data represents student opinions from across the state?  This would require a better understanding of population and samples, instruction on sampling methods, and a better foundation in randomness (all of which are where I would insert my videos, a lecture, or a mini activity to instruct on these needed stills).
• How effective was the presentation of the data?  Did you share everything or select only the most interesting graphs?  Did you break any graphs down based on conditions or only look at overall answers?  Did the audience find it visually appealing and interesting to look at?  Does the presentation of the data deceive or dance around any issues?  What other ways could your data be presented visually?
• How well did you communicate the uncertainty in your results?  Does 75% mean exactly 3 of 4 or is there a margin of error?  Where does that margin of error come from?  Does it account for bias in question wording, questioning technique, or sampling method?  Could you find a confidence interval on your first survey data, and why / why not?  Can you explain in only a few characters on an infographic what your confidence interval tells you?

This would lead to a new (or simply refined) survey project where students generate a set of 5-8 questions that they find interesting, limiting themselves primarily to 2-option multiple choice questions (would-you-rather questions work great for this) and quantitative response questions.  Scale questions (rate from 1-5...) are technically not quantitative because the difference between 1 and 2 may not mean the same as the gap from 2 to 3, plus the numbers are usually more subjective, but they work well (somebody please convince me they are okay or not okay).  Otherwise counting how many days/week or times/day a person does something works too.

To collect the data, we would probably use a school-wide random sample stratified by grade (gender would be nice too, but it is harder based on the lists I get access to).  Instead of having many groups annoy every classroom, we would try to consolidate all of our questions into one or two surveys and split up the randomly selected students.  We would only decide this after class discussion and consensus, so they might choose a different method if it works well (like regular SRS).

With data in hand, I plan to teach mini-lessons on data presentation, learning primarily by example and discussion.  We would start with traditional graphs like stem plots, box plots, histograms, dot plots, bar graphs, and pie graphs, making sure we use the right ones for the type of data / size of data set, and discussing how they could be extended or improved to be more visually appealing (such as making the bars some kind of image or filling in a proportion of an image instead of a pie graph).  To push students beyond the basics and force them to think more creatively, this would definitely include my favorite graphics of all time: Death and Taxes and Napolean's March (both below, click to enlarge if you have a ton of time to kill, because if you read my blog, you're nerdy enough to stare at them for hours).  

 

In addition to visual summaries, we like to summarize data in numbers.  Categorical data gets boiled down to a proportion while quantitative data gets a lot more complex as you describe multiple measures of center and spread and consider both in the context of its shape.  Discussing the tradeoffs between mean and median and using case studies where students defend which statistic is the better summary could set up a nice baseline before they start throwing numbers all over their infographics.

Before students get too carried away with how they will present their graphs and summary stats, we need to spend time learning about the limits of our precision.  Connecting back to sampling, we can simulate in class the idea of taking an SRS of the same population multiple times with varying results, eventually getting to the big question -- is the average and variation predictable?  After a few time-consuming in-class simulations, we can take a trust leap into a computer simulation (I like StatKey).  Thousands of iterations reveal that there is in fact pattern and order to the means of the simple random samples, allowing us to say how confident we are that any one sample fell within a certain range of the true population mean.  I plan to get deeper into inference in the next unit, so just introducing the idea that uncertainty is predictable should be mind-blowing enough for now.

As we work through all of these components in class, there will be a few individual assignments and checkpoint quizzes (probably multiple choice for instant grading and feedback) to make sure all students are learning the core material.  If a student does not score 80% or better on the quiz, they will need to meet with me individually or in a small group before their group can turn in any shared assignments (thanks to Kris @KHaeussinger for that idea).  This adds individual accountability and a check that the supporting content is being taken seriously and making sense.  Beyond that, students will spend most of their time working on their infographic project in teams of 3 that I create at the start of the unit (based on experience as a student and as a teacher, 3 just seems like a magic size for most things).  As we get about halfway through the unit, I want teams to start to give each other feedback.  There are structured formats for this such as a gallery walk or critical friends that should be used to quickly generate constructive ideas instead of "oh yeah that looks great".

The final product is a large color poster for display in the halls of the school.  To share them with the class, we would hold a poster session where one teammate stays back and two float to see other posters and listen to mini-presentations.  Rotate 3 times and everyone would both present and watch.  I would float and ask questions / assess the final product (pulling in the floating team members when I'm present at their poster).

WHEW.  That was a lot longer than I expected...I guess I had been giving it some subconscious thought for a while.  That said, none of this is set in stone and I want it to be awesome, so please point out the areas that could be stronger or got overlooked and I look forward to incorporating your feedback.  Shoutouts to Megan @veganmathbeagle and Dianna @d_hazelton for helping me think through some of this and pushing me to write it down.  Besides ideas for seniors, it will be awesome to hear what this might look like for 9th graders / non-seniors as well.

A half-baked unit idea for hypothesis testing

The first topic that broke me as a teacher was teaching hypothesis testing as a student teacher.  It broke me so bad that I totally changed my stats course to better engage students in the statistical process.  It was the catalyst that fixed so many other things in my class, and did improve how I teach hypothesis testing, but kids still struggle with it more than anything else.

At TMC14 last week, one huge theme in many of the sessions and "My Favorites" was getting students to talk about math.  The example that stood out to me was Chris Luzniak (@PIspeak)'s first session on debating in math class.  He focused on structured arguments.  When students raised their hand to share, he asked them to make a claim (statement) and then give a warrant (reason).  He builds a culture of listening, clearly stating a view, and backing up that view with sound logic.  Chris uses mostly informal debate (more like a classroom discussion), but for the purposes of this new unit I think I want it to be more of a prepared debate.

I am trying to envision a unit where hypothesis testing is taught in multiple ways and students, as their unit project, are left in teams to defend their preferred method for making decisions with data.  I see two big questions that should be very approachable by my students:.  The first is whether to use simulation or a probability model to obtain a p-value:

1. The AP Statistics curriculum, along with nearly every other stats course, uses a probability model (the normal curve) as the basis for inference.  Because of the Central Limit Theorem, even the craziest of distributions have a very normal looking sampling distribution if the sample size is large enough (often >30).  This method for doing inference is clean and analytical (could be worked out by hand).  The downside is that, like all models, there are some assumptions that must be made that are not always valid, especially with smaller samples.
2. The preferred method of many programmers (including me) is to create a very simplistic model in software and repeat it thousands of times to create a sampling distribution.  This method is called bootstrapping.  Even though programming may not be accessible to most students, tools with clean interfaces such as StatKey make it easy to perform the calculations.  This method for inference is very easy to understand and explain (drawing values from a hat), and since computers are fast, it is just repeated many thousands of times.  It also does not require any assumptions about the size or shape of the sample's distribution.
3. If you understand Bayesian hypothesis testing, you could have kids argue for the need to use likelihood ratios based on meaningful prior probabilities.  Despite spending half of the day researching it, I really don't understand it well enough to explain it to anyone else, so this option is out for me.  If you want to learn, one awesome option is a free book one of my college computing professors wrote called "Think Bayes".

The second question is why we need p-values at all:

1. The traditional hypothesis test starts with a default claim and seeks evidence that suggests it to be unlikely.  The statistician starts with a threshold level, such as 5%, and tries to show that there is less than that probability of finding the data they found when he assumes the null hypothesis to be true.  The actual probability, the p-value, is often reported along with the decision to reject or fail to reject the null.  This is seen in nearly all science publications.
2. Another viewpoint held by a minority group of stats folks (including me) is to not use p-values at all.  Instead, a correctly computed confidence interval could be compared to a null hypothesis to see if the null mean is captured by the interval or not.  For example, a soda pop can of Coke (covered all my regions there...) claims to have 12oz of liquid inside.  I am trying to prove that it is less than 12oz in a one-sided test at the threshold of 5%.  Instead of finding a p-value, I could create a confidence interval around my data.  Since I want each tail of the interval to have 5%, the interval would be the middle 90%.  If my 90% confidence interval was 11.5oz to 11.9oz, then I could conclude that I was being ripped off.  The reason I like this approach, despite its apparent complexity, is that it doesn't just tell you that you are getting ripped off -- it tells you how much pop you can expect to get in the can.

The point is not for the teacher to ride their high horse about their answers to each question.  The fact is that there is a group of intelligent people who support all of these answers, so they could all be defended in front of the class.  In order to make a good defense, each group needs to understand their opponent's responses well enough to counter their argument.  The normal-curve based model will be fairly non-intuitive to my students since I plan to introduce them to inference using bootstrapped confidence intervals in a unit on infographics.  They will need to understand enough about the Central Limit Theorem to argue why the normal curve is a good model and know the assumptions that are made to use it.  They will want to demonstrate how easy calculation can be with a TI-83 and how easy it is to explain with a sketch of the normal curve.  The simulator groups will need to understand when every simulation yields a different answer and why increasing the number of bootstrapped samples converges the answer to a more precise value.  They need to understand why they don't need to check any assumptions (other than having a SRS) like the other groups need to do.  There are a similarly large number of key ideas that need to be well understood about the second question to properly debate it as well.

The big graded tasks in the unit would probably be giving and receiving feedback in mini-debates between paired teams, a final debate between groups in front of the entire class, and a write-up after the debates picking a personal stance and communicating all that they learned in the process.

The unit would be kicked off with the big question of "how can you use data to make decisions"?  From there, we would generate a list of topics we needed to know more about in order to get ready for the debate.  Some of these big ideas I would lecture on for the class, while less conceptual ideas would be left for students to watch in my videos.  I would also give short quizzes to make sure students were grasping the basics of hypothesis testing and the different approaches so I could target struggling students and their groups for quicker intervention.

So...I need help.  I'm sure there are a ton of holes in this concept or things I did not clarify that really matter.  I would love to not only turn this into an awesome unit for my classes, but for everyone who would want to use it, so please poke holes as ruthlessly as if it were your own curriculum.  Thanks!