Showing posts with label Probability and Statistics. Show all posts
Showing posts with label Probability and Statistics. Show all posts

Friday, May 17, 2013

Day 14 - EQAO and Equity

The Problem...

What educational equity issues become apparent when looking at EQAO results?

Why?
  • Provides relevant context for statistics concepts
  • Opportunity to turn a social justice lens on inequity in education
  • Potential for Grade 12 students to help Grade 9 students get prepared for EQAO testing
This idea comes from a lesson example that I developed for a curriculum interrogation assignment at OISE.  It's an interesting lesson idea that I thought would be good to improve by looking at it through a 21st Century lens.  

The Math Curriculum...

I have stated before that the math curriculum is very concept-heavy (at least compared to Science which distinguishes STSE and skills expectations from concepts).  This generally leaves little room for deep investigations about culturally relevant and engaging topics.  

The vast majority of examples of using social justice education, global citizenship education, culturally relevant/responsive pedagogy or critical pedagogy in math class that I have heard have been based on statistics.  Unfortunately, in the high school math curriculum, statistics really only seriously comes into play in Grade 12 Data Management.  I think one way of increasing student engagement in math classes would be to spread out the Data Management concepts throughout the grades (another may be providing teachers with a bit more flexibility with the concepts).

Because Data Management is the obvious math course for the delivery of social justice education, I have stayed away from it in my blog.  Up until this point, my math entries have focused more on being critical of bias in math in general.  This however is an investigation about how we can use math to look at equity in society.  It fits in directly with a bunch of the Data Management expectations.

EQAO and Equity...

The following plot compares Primary EQAO data to Stats Canada Data on average family income.  Each dot represents a school's average result.  The article this plot comes from can be found here (it was published by EQAO in 2008).  This plot alone can be the starting point for a serious discussion on equity in standardized testing and education. 

I thought a good way to start the class would be to hand out a bunch of questions from a past Grade 9 EQAO test.  This will get them in the frame of mind of a Grade 9 student.  They can start to think about what factors may give students students trouble with those questions.  Note: A bunch of past secondary EQAO tests and other EQAO related resources can be found here.  

Find the relationships...

Then they could start looking at data.  The above plot can be found online but to facilitate Knowledge Construction students should be going to the EQAO website to get the raw data.  They can then draw their own relationships.  Some of the simple relationships they can get directly from the EQAO data are gender and stream comparisons.  Questions that may be good discussion starters are:
  • does the division between genders seem to be increasing or decreasing over time?
  • does the division between academic/applied seem to be increasing or decreasing over time?

Other sources of data students can compare the EQAO results to are Stats Canada data and even the Fraser Institute Rankings (which frankly scare me because of how seriously they seem to be taken).

Critical analysis...

After establishing relationships, students can begin discussing why some of the relationships exist.  As a teacher, this is where things can become difficult and exciting.  The discussion is the part of the investigation that can be most thought provoking and interesting but often gets left out for the sake of saving time.  I believe we're talking about 21st Century Fluencies, that excuse is unacceptable.  Bypassing the discussion doesn't help students develop the Skilled Communication and Real-World Problem Solving skills necessary for the 21st Century Learner.

I think another huge topic of discussion is taking a step back and looking at standardized testing in general.  What is the purpose of EQAO?  Really it is to evaluate teachers and schools.  Is this the best way to do this?  Is it a good representation of the academic strength of the school?  How else should schools be evaluated?  What equity issues may arise because of standardized testing?  What biases are present in both the test itself and the data it produces?  

It would also be interesting to look at the standardized testing situation in the United States, where it is much more pervasive and compare it to Ontario.

Action!

Now, how can these results be used by the Grade 12 Data Management Students?  I would say it's a good opportunity for the Grade 12 students to help the Grade 9 students that are preparing for the EQAO.  If they find relationships in the data that could be beneficial to the younger students then they could share their results.  Most likely, they will not be able to directly influence any of the factors that cause inequity but they can at least advocate for and educate the Grade 9's.  Most students probably never think back to EQAO after they write it but I think this is a good opportunity to build some school community and leadership skills by having Grade 12 students setting up tutoring sessions for Grade 9's.  There's the Collaboration!

21C...


1. Collaboration: entry - adoption - adaptation - infusion - transformation
2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
3. Real-World Problem Solving & Innovation: entry - adoption - adaptation - infusion - transformation
4. Skilled Communication: entry - adoption - adaptation - infusion - transformation

Tuesday, May 14, 2013

Day 11 - Hockey Stats

The Problem...
  • How are all those hockey stats collected and what do people do with them?

Why?
  • Engage students with the data collection and management behind sports
  • Expose students to the real-world application of statistics
  • Entry point to talking about the cultural impact of hockey in Canada
This entry is coming a day late because I was too upset to write one after the Leafs loss to the Bruins in overtime in Game 7 yesterday.  It's fresh on the minds of Leafs fans so I decided I would blog about it and somehow tie it to math.  Below is an nhl.com screenshot of the top 10 players from the 2012-2013 NHL regular season:

http://www.nhl.com/ice/playerstats.htm?season=20122013&gameType=2&team=&position=S&country=&status=&viewName=summary#?navid=nav-sts-indiv

That is a lot of stats to collect for each player!  In addition to analyzing the data on the site in a Grade 12 Data Management class, the question of how and why they collect these stats at all would be an interesting way of pulling the topic into the 21st Century Classroom.  

Close to home...

I think an awesome activity would be for a class to organize in order to collect an array of statistics of a team's season.  Ideally it would be a season played by their own high-school team so small groups of students could go out to collect stats for each game.  The class could consult with the coach/team at the beginning of the year in order to Collaboratively decide what statistics would be most beneficial for the team. They could analyze the stats over the course of the year and create reports to the coach who could ideally use them to improve the team's performance, supporting both Real-World Problem Solving and Skilled Communication.

In addition to potentially providing the coach with tools for improving the team's performance, the purpose of this activity would be to expose students to the process for the collection of data and some of the complications that may arise.  This Knowledge Construction would not occur if students were just studying stats taken from nhl.com.

Note: this obviously doesn't have to be a for a hockey team.  It could be any sport. Hockey is just what I had on my mind at the time of writing.

Digging deeper...

Once they have collected the data, they could start asking the deeper questions about collecting data in sports.  For example: in what ways are the data collected in hockey games are used?  Below are some things that may come up:
  • Help teams improve their performance:  find ways to improve their team or find weaknesses in other teams
  • Scouting for the NHL: the site behindthenet has a comprehensive report on how junior player's stats are projected to estimate what their performance in the NHL would be
  • Fans love stats!  Stats get fans more engaged.  It is a source of revenue for the teams and leagues.
  • Provide research for medical studies (for example: the concussion epidemic)
Here are some other questions that may be good discussion starters: 
  • How much money is spent to collect stats in the NHL?
  • Is it worth all the money to collect the stats?
  • How many people does it take to collect stats for one NHL game?
  • How might projecting a Junior players stats into the NHL be problematic?
21C...

1. Collaboration: entry - adoption - adaptation - infusion - transformation
2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
3. Real-World Problem Solving & Innovation: entry - adoption - adaptation - infusion - transformation
4. Skilled Communication: entry - adoption - adaptation - infusion - transformation

Future lesson ideas...
  • Along the same lines but for Baseball: the math behind Moneyball (the movie)

Thursday, May 9, 2013

Day 8 - The Odds

The Problem...

What are the odds of winning the jackpot on a slot machine?

Why?

  • Provides context for probability and combinatorics topics
  • Engage students in discussion on ethics of gambling and the Toronto Casino Debate


Being in Niagara Falls yesterday got me thinking about casinos and gambling, an interesting topic for debate and a common way of illustrating concepts in probability.  

Also, the debate on whether or not Toronto should allow a casino to built or not provides context for a discussion of the topic in a classroom setting.



My internship supervisor and I were wondering what the actual odds of winning at a slot machine were.  What does an 80% payout mean?  It turns out payouts are deceiving.  A 80% payout doesn't mean you will win 80% of the time but the value you can expect to walk away with is 80% of what you started with.  In probability, an expected value of 100% (or 1) is considered a fair game because the player and the casino have equal chances of winning.  

According to this website, Nevada law states that the minimum payout can be 75%.  It is rarely this low however because casinos compete for players by increasing the payout.  90-95% are more common.  In addition to the games being mathematically unfair, they are built so that it is more likely that the machine will stop in the blank spaces on either side of the jackpot symbol to make it feel like the player is close to winning.

The Curriculum...

As an activity in class, students could calculate the odds of winning a jackpot on a slot machine with (for example) 3 reels and 20 positions (symbols + spaces in between symbols) on each reel.

Then, questions get more complicated when trying to find out the expected value (or payout) of a machine based on the probabilities of landing on each symbol and the payout of each symbol.  This website provides the background for some of that math.

These questions touch on various concepts central to the Grade 12 Data Management course (probability, random variables, combinations, expected values, probability distributions).


21C...

As an extension and problem-solving exercise would be to devise a slot machine that would provide a fair (100%) payout.  Students with experienced in programming could write a program to simulate the machine.

A lot of the time in Data Management, gambling and game related probability problems are given to students.  I think it is important to put these questions in context by critically thinking about and discussing the ethics and psychology of gambling.  The Toronto casino debate is potentially an entry point for the topic.


2. Knowledge Construction: entry - adoption - adaptation - infusion - transformation
3. Real-World Problem Solving & Innovation: entry - adoption - adaptation - infusion - transformation