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Baseball, Brain Cancer and Relative Risk

The 1993 Phillies were the most fun team I have ever followed.  I was nearing the end of my college years, and I vividly remember the insane night when me and 3 buddies celebrated a win at 4:30 in the morning, and the exact spot I was sitting when our hearts were broken with Joe Carter’s home run (I still look away when it comes up on highlight reels).

This week the catcher of that team, Darren Daulton, died after a battle with brain cancer.  Newspapers have shared memories of “Dutch” and among the articles is one which reminds us of the surprising number of former Phillies who have passed away due to brain cancer (Tug McGraw, John Vukovich, and Johnny Oates). A revised 2013 article from the Philadelphia Inquirer analyzes the the unusual number of Phillies who have developed brain cancer, and contains many appropriate entry points for Statistics courses.  Some highlights from the article:

  • A comparison of the observed effect to random chance – here a professor of epidemiology summarizes: “You can’t rule out the possibility that it’s random bad luck.”
  • A summary of plausible variables which could lead to elevated levels of exposure, such as artificial turf (which may have contained lead) or anabolic steroids.
  • An analysis of the increase rate of brain cancer among Phillies – here we are told that the Phillies’ rate is “about 3.1 times as high” as the national rate.  A confidence interval, along with an interpretation and associated cautions are also included.

Let’s explore that “3.1 times” statistic…time to break out the technology.


A few weeks back, I attended the BAPS (Beyond AP Statistics) workshop in Baltimore, as part of the Joint Statistics Meetings. Allan Rossman and Beth Chance shared ideas on using their applet collection to explore simulation (see my earlier post using the applets to Sample Stars) along with a “new” statistic we don’t often talk about in AP Stats – relative risk.

To start, I used the Analyzing 2-Way Tables applet and used the “sample data” feature.  Here I attempted to use the same numbers quoted in the article:

The national rate was 9.8 cases per 100,000 adult males per year, while the rate in the former Phillies was 30.1 cases per 100,000 – about 3.1 times as high.

There are two issues here: first, to perform a simulation we need counts, so numbers like 9.9 and 30.1 just don’t play nice.  I’ll use 10 and 30.  Also, I wasn’t surprised that this site was not real happy with my using a population of 100,000 for simulation.  Here, I am going with 1,000 for convenience and to make the computer processor gods happy – we can debate the appropriateness of this down the road.

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The applet will then simulate the random assignment of the 2,000 subjects here to the two treatment groups (group A: being a Phillie, group B: not being a Phillie). How likely is it that we will observe 30 or more “successes” (which here represent those who develop brain cancer) in one of the two groups?  In the applet, we can see how the “successes” have been randomly assigned from their original spots in the 2-way table to new groupings.

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AT BAPS, Allan Rossman then explained how we can summarize these two groups using Relative Risk, which is listed under the “Statistic” menu on the applet. In general, relative risk is the proportion of success in one group divided by the proportion of success in a second group.  If we have proportions in two groups which are equal, then the relative risk would be 1.  We can then link to the newspaper article which claims a 3.1 “relative risk”, simulate many times with the applet (below we see the results of 10,001 simulations), and compare to the reported statistic.

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According to the simulation, we should only expect to see a relative risk of 3 or above about 0.08% of the time – clearly an “unusual” result.

But the article does not claim a significant difference, and cautioned against doing so as a number of assumptions were made which could alter conditions.  This would be an opportunity to discuss some of these design assumptions and how they could change the outcome.

Rest in Peace Dutch!

 

Area Models and Completing the Square

I’m nearing the end of my time with my 9th graders, with this week dedicated to moving beyond factoring as the sole method for solving quadratic equations and towards more general methods like completing the square.

Late in May, David Wees shared materials which challenge students to investigate the relationships between “standard form” and “completing the square” form (aside – does anyone agree on proper terms for these?) using area models to build representations.  Given that I use area models often to introduce polynomial multiplication, I was eager to maintain consistency in the student understanding.

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But before we dove into David’s lesson, I wanted students to revisit their understanding of area models.  In this Desmos Activity Builder lesson I created, students shared their interpretations of area models and worked in pairs to investigate non-square models.  In one of the final screens, students argued for the “correct” interpretation of a model.

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Using the Desmos teacher dashboard, we could see clear visual arguments for both representations.  This was valuable as we ended the lesson for the day, and tucked that nugget away for Monday, when we would begin to formalize these equivalencies.

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After the weekend, students worked independently through David’s Completing the Square lesson. Not only did students quickly move through the area models and the dual representations, the debates between students to explain how to move from one representation to the other were loud and pervasive.  I’m also loving how many of my students have started to use color as an effective tool in our OneNote-taking (below).

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At the end of the sheet, all students completed problems which translate standard form to vertex form with no support from me (“no fuss…no muss”).  It dawned on me that something amazing had happened….my students had figured out completing the square without my ever talking about completing the square.

Tomorrow we’ll tackle those pesky odd-number “b” terms, but my students own this already!

Another Pascal’s Triangle Gem

At a recent local conference, Jim Rubillo, Annie Fetter and I were saying our good-byes at the end of a fun evening, when Jim’s puzzly side emerged…

What proportion of the numbers in Pascal’s Triangle are even?

Every time I talk to Jim, he’s bound to have a neat problem for me to chew on.  The last time, he shared a fun task involving the harmonic series. Take a few minutes and think about this Pascal’s Triangle scenario…I’ll even leave you some spoiler space.


Jim RubilloAt the ATMOPAV Spring Conference last month, Jim shared an entertaining talk titled “Gambling, Risk, Alcohol, Poisons and Manure – an Unfinished Life Story”.  The talk led the attendees on a journey through the history of statistics, starting with games of chance and the meeting of Chevelier du Mere and Blaise Pascal, through the introduction of formal inference procedures developed at the Guinness brewery, and to identifying statistical abuses in the present day.

Jim is a life-ling educator and former Executive Director of NCTM who happens to live quite close to me.  It was a thrill having him share his ideas with the group.


Had enough time to think about this Pascal question?  Spoiler time is up!

So, which rows are in Pascal’s Triangle are we talking about here?

In theory, we are talking about “all” of the rows in the infinite Pascal’s Triangle, which makes this a bit tricky to think about for kids (and adults as well!).  But Jim shared with me slides which show the proportion of evens in increasing numbers of rows of the triangle.  You will notice that as the number of rows grows, the proportion of even entries also increases, and approaches 1.  What a neat result!  Below is an animated gif I made using a Pascal’s coloring applet which shows the increase in the proportion of even (white space) numbers in increasing rows.

Pascal Zoom

For your class, this is a fun opportunity to talk about the parallels between Pascal’s Triangle, Sierpinski’s Gasket, and fractal area.

Already looking forward to my next encounter with Jim!

Drinking the Statistical Power Kool-Aid

For my colleagues who teach AP Stats, there are few phrases more terrifying:

Today I am teaching Power.

Power: a deep statistical concept, but one which often gets moved towards the back of the AP Stats junk drawer.  The only mention of power in the AP Stats course description comes under Tests of Significance:

Logic of significance testing, null and alternative hypotheses; p-values; one- and two-sided tests; concepts of Type I and Type II errors; concept of power

So, students need to understand the concept of power, but not actually compute it (which is itself not an easy task).  Floyd Bullard’s article “On Power” from the AP Central website provides solid starting points for teachers struggling with this concept; specifically, I appreciate his many ways of considering power:

  • Power is the probability of rejecting the null hypothesis when in fact it is false.
  • Power is the probability of making a correct decision (to reject the null hypothesis) when the null hypothesis is false.
  • Power is the probability that a test of significance will pick up on an effect that is present.
  • Power is the probability that a test of significance will detect a deviation from the null hypothesis, should such a deviation exist.
  • Power is the probability of avoiding a Type II error.

This year, I tried an activity which used the third bullet above, picking up on effects, as a basis for making decisions.

HEY KOOL-AID MAN!

Photo Mar 06, 12 11 56 PMArriving at school early, I got to work making 3 batches of Kool Aid.  During class, all students would receive samples of the 3 juices to try.  Students were not told about the task beforehand, or where this was headed. Up to now, we had discussed type I and type II error, so this served as a transition to the next idea.

THE BASELINE SAMPLE:

All students received cups and as they worked on a practice problem I circulated, serving tasty Kool Aid – don’t forget to tip your server!  I told students to savor the juice, but to pay attention: I promised them that this first batch was made using strict Kool Aid instructions.  Think about the taste of the juice.

SAMPLE A:

Next, students received a drink from “Sample A”.  Their job – to assess if this new sample was made using LESS drink mix than the baseline batch.  Also, I varied the amounts of juice students received: while some students were poured full cups, some received just a few dribbles.  To collect responses, all students approached the board to contribute a point to a Sample A scatterplot, using the following criteria:

Photo Mar 06, 8 50 05 AM - CopySample size: how much juice you were given

Evidence: how much evidence do you feel you have to support our alternate hypothesis – that Sample A was made with LESS mix than the baseline?

As you can see, the responses were all over the place – a mixture of “we’re not quite sure” to “these are strange directions” to “I just don’t trust Lochel – something’s up”.  But the table has been set for the next sample.

Sample A: it was made with just a smidge less mix than the baseline.  So I wasn’t totally surprised to see dots all over.

SAMPLE B:

Photo Mar 06, 8 50 05 AMI poured drinks again from this new sample, and again varied the sample sizes.  I asked all students to think about their evidence in favor of the alternate, and wait until everyone tasted their juice before submitting a dot.

And check out those results!  Except for a few kids (who admitted they stink at telling apart tastes), we have universal support in favor of the alternate hypothesis.

Sample B: this was made with 1/2 the suggested amount of drink mix.  Much weaker!

FOLLOW-UP DISCUSSION:

This activity made the discussion of power much more natural.  In particular, what could occur during a study which would make it more likely to reject the null hypothesis, if it deserves rejecting?

Larger sample size: smaller samples make it tough to detect differences

Effect size: how far away from the null is the “truth”.  If the “truth” as just a bit less than the null, it could be difficult to detect this effect.

In terms of AP Stats “concepts of power”, this covers much of what we need.  Next, I used an applet to walk students through examples and show power as a probability.  And like most years, this was met with googly eyes by many, but the foundation of conditions which would be ripe for rejecting the null was built, and I was happy with this day!

Suggested reading: Statistics Done Wrong by Alex Reinhart contains compelling, clear examples for teachers who look to lead discussions regarding P-value and Power.  I recommend it highly!

4 Activity Builder Formative Assessment Ideas

The creative team at Desmos continue to develop engaging lessons using their Activity Builder interface, found at teacher.desmos.com. While teachers I encounter have their own favorite activity, many desire to dive in and create their own. But building your own activity, testing it, and hoping it works with your class can be an intimidating task (pro-tip: making your own activity is really hard!).  But there are a few simple ways teachers can use Activity Builder as a mechanism for formative assessment.  Here. I share 4 quick and easy ideas – you can check them out and observe their structure at this link.

SELF-CHECKING GRAPH MATCHING

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I used this often with my Pre-Calculus class in the fall, and the concept works equally well with younger students.  Simply start a new Activity Builder screen, and enter the equation you’d like students to provide.  Place the equation in a folder, which you can hide so students won’t see it when they encounter the screen.  Finally, by making the graph with dashed lines, students can easily see if their submission matches the requested graph, and can adjust accordingly.

GALLERY WALK

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Here’s a neat Activity Builder hack you may not know about.  If you have an existing Desmos graph, copy the URL from your graph to the clipboard.  Then, in an Activity Builder screen click the “Graph” button and paste the URL into the first expression line – and PRESTO, the graph is imported into an Activity Builder screen.  I often collect student work by simply having them submit a Desmos URL.  Consider taking samples of student works and create a virtual gallery walk.  Let students view each other’s ideas, comment and make suggestions. Thanks to my colleague DJ for providing neat student graphs!

SELF-ASSESSMENT SLIDERS…AND OVERLAY

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Have students assess their own learning with a moveable point. Provide an “I can…” prompt and let students consider where they fall in the learning progression.  Hold a class-wide discussion of unit skills by anonymizing student names and using the overlay feature to take the class pulse on skills.

MY FAVORITE DISTRACTOR

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Activity Builder allows teachers to build their own multiple-choice questions, with the option of having students provide an explanation for the choice they make.  In “My Favorite Distractor”, students select an answer they KNOW is wrong, and explain how they know.  This may not work for many multiple-choice type questions, but consider using this idea in situations where the distractors have clear, interesting rationales for elimination.

Have your own quick formative assessment ideas?  Share it here!

 

A Bulleted Assemblage of Items for the New School Year (but not a list)

The “list” article is a popular device, and one which often draws the eyeballs. Lists are also, often, a cop-out – a way to express many ideas without having to dig too deeply.  I hate lists….

As I start my new school year tomorrow, I give you this bulleted assemblage of items which are on my mind as I look forward to our first day.

  • Fawn Nguyen’s 7 Deadly Sins of Teaching Math is required reading for all professionals. In particular, I strive to pay more attention to my (teacher talking / student talking ) ratio.  I like to think I am strong in this area, but I need to do better. Before the end of the last school year, our district screened the movie “Most Likely to Succeed” to all professional staff.  In an opening scene, the teacher provides first-day freshmen with an opening day task – and then leaves the room.  The students struggle, the teacher eventually intervenes, but a powerful classroom culture is established.  I want to provide more tasks to my students where I’m simply not needed.
  • I have used a number of opening-day activities for AP Statistics over 14 years. Distracted Driving and the Henrico hiring case are two I used most often. But I think Doug Tyson’s Smelling Parkinson’s activity could be my new favorite. It’s a powerful premise which gets kids talking about the possible vs the plausible on day 1, with a hint of simulation thrown in for good measure. I show the video below to the class and right away the statistical importance of what we do for the entire school year is established.
  • Desmos Activity Builder will take on a much bigger role in my classroom.  I’ve created activities for both my Pre-Calculus and my freshman Prob/Stat class to review their understanding, and also to serve as my “getting to know you” opportunity.  Look forward to sharing out how it goes.
  • Shoes.  I hate new shoes. They’re tight and often rip apart the back of my ankle until I break them in.  If we can have pre-washed jeans, then we can have pre-worn shoes.  We need our best people on this.
  • Who knew a cute Pythagorean triple generator could be of interest to so many. After I posted about an interesting share from Ken Sullins at the PCTM summer conference, so many folks chimed in with their ideas.  Thanks especially to Joel Bezaire who shared additional ideas from Twitter Math Camp.  I’m using this in my pre-calc class on day 1.

  • I’ve given the same probability problem to my freshmen for the last few years. I love everything about this problem on day 1: it gets kids talking, it gets kids struggling, and it tells me much about their problem solving background.

 

OK, maybe this was a list after all.  I need to do some last-minute ironing.

The People in My Math Neighborhood

Oh, who are the people in your neighborhood?
In your neighborhood?
In your neighborhood?
Say, who are the people in your neighborhood?
The people that you meet each day

I work with an awesome group of people at a high school outside of Philadelphia.  They are my colleagues, the people I share ideas with on a daily basis, and some of my closest friends.

But in recent years, my math neighborhood has grown considerably.  I suppose I discovered the power of the online neighborhood 4 or 5 years ago, developing and growing a wonderful network of professional colleagues through the #MTBoS. And my relationship with this neighborhood has grown from a mechanism for sharing ideas, to a source of inspiration, positive thought, discussion and reflection.

We are now 3 weeks after the NCTM Annual Conference in San Francisco. It’s easy to forgot the little things which occur in a big conference, and I hopefully will find time to reflect and utilize new ideas later. NCTM this year has done a wonderful job of providing a means to continue reflections and growth outside of the conference, along with archiving session resources.  Here, I highlight 4 sources of inspiration, and friends in my math neighborhood, as I look back on my San Francisco experiences.

GRAHAM FLETCHER – Graham, an elementary specialist from Georgia (or is he Canadian? such a chameleon), challenged teachers to consider the mathematical story we share with students in his ShadowCon talk. How is your story different than the one being told by your colleague teaching the same material just across the hall?

grahamHigh school teachers may be intrigued by Graham’s discussion of fractions, reducing and equivalence and the role of “simplifying”. His talk has caused me to think about the many odd restrictions we place on student work: i.e. “write the equation of your line in standard form”, and their necessity in my math story. Graham’s call to action – challenging teachers to identify their own “simplifying fractions” (something they teach not currently in the standards) – is an appropriate task for all grades.

ROBERT KAPLINKSY – Robert was featured on the MathEd Out podcast last summer, and I recall taking a walk, listening last year when it occurred to me that Robert’s path to becoming a math teacher was eerily similar to mine. His ShadowCon talk, “Empower”, reminds me that no matter how top-down our education world may feel, we all have a role to empower others and become influential in our math neighborhoods. I appreciate the multiple mechanisms Robert suggests for fostering empowerment, and his call to action that we thank a colleague who helped us feel empowered is a wonderful way to close out a school year – and look forward to new things.

PEG CAGLE – I have admired Peg’s ideas for some time now, and was thrilled to meet and chat with her last year at Twitter Math Camp. Even though I rarely teach geometry, I felt pulled to Peg’s session “Paper Cup + Gust of Wind”, and was awed by the simplicity, engagement, theme-building in this simple task. By rolling a paper Dixie cup along a surface, Peg develops a lesson which extends through the school year, building complexity each time.

Day 1:Explain what happens when we roll out the cup

Day 40: Convince a skeptic of the shape it makes. Find its area.

Day 105: find area of shape based on dimensions

Day 140: How can you build a cup from a single sheet (with base) of 8.5 x 11 paper to trace out the maximum area as it rolls?

Day 175 (after trig ratios): how do you NOW find the area of the shape, given its dimensions

This session has caused me to think about other simple tasks which could become full-course themes. Peg’s inspiration came from a cup blowing in the breeze – you never know where the next fun math idea will come from!

CHRISTINE FRANKLIN – Why was I so nervous and awe-struck to meet Christine at the AP Stats forum in San Francisco? Because she is so awesome – and was the inspiration for my NCTM talk on Variability and Inference, geared towards the middle school community. It was at Professional Night at the AP Stats reading 2 years ago where Christine diagrammed the historical path stats has taken in K-12 curriculum, and the parallels between AP and middle school descriptions. Christine was recently named the K-12 statistical ambassador by the ASA, and a sweeter person could not fill the job.

Hoping I never move out of the neighborhood!