Common Core Anyone?

There has been a great deal of discussion in public media about the Common Core State Standards.  We at Westminster have developed our own Essential Learnings for all our courses, but are keenly aware of what the Standards say.  I believe that at the center of them is the need for some kind of common national mathematical curriculum that serve as a guideline for local curriculums.  That at least as how we view them and use them in evaluating whether our curriculum meets the needs for our students.

I do think that no matter the curriculum taught, the way in which it is taught becomes far more important.  I do like the Standards for Mathematical Practices and am striving to include more of them in my daily instruction.  Here they are:

   Make sense of problems and persevere in solving them. (MP1)
   Reason abstractly and quantitatively. (MP2)
   Construct viable arguments and critique the reasoning of others. (MP3)
   Model with mathematics. (MP4)
   Use appropriate tools strategically. (MP5)
   Attend to precision. (MP6)
   Look for and make use of structure. (MP7)
   Look for and express regularity in repeated reasoning. (MP8)

These are each worthy practices that encourage student growth in their ability to be successful with mathematics and to appreciate its beauty.  The challenge is how to get students to grow and understand where they currently are with their thinking.  Jill Gough and Jennifer Wilson have begun developing learning progressions to help students see where they are with each of the Practices and what they need to do to reach the Level 3 target.  Over the next few blogs, I will be reflecting more on each of the Math Practices and how their use is impacting my class.


The Mathematics of Fantasy Sports – NFL History Books

The class’ next task was to play an NFL Fantasy season, using data from a past season to predict the next season.  NFL fantasy football was by far the game that more students played.  Here is the lesson plan–

  • The class will be divided into 8 teams (3 teams of 3 and 5 teams of 2).
  • Each team will research how various Websites (such as ESPN, Yahoo or CBSSPORTs) allocate points for a fantasy NFL Team that consistes of 8 positions: 1 QB, 2 RB, 2 WR, 1 TE, 1 K and 1 DF/ST.
  • The class will determine how the points will be allocoted for the play of 8 NFL positions: 1 QB, 2 RB, 2 WR, 1 TE, 1 K, 1 Flex and 1 DF/ST.
  • An Excel spreadsheet will be created by each team to track their team’s progress.
  • A random drawing will occur for the draft order.  A serpentine draft will be used for the draft – round 1 will be teams 1 – 8 then round 2 will be teams 8 – 1 and  so forth.
  • The teams will then research players from the 1988 NFL season and discuss draft strategy before choosing a team. YOU MAY ONLY LOOK AT SEASONS 1988 AND BEFORE. Here is a link:1988 NFL Season Data
  • Here is an Excel Spreadsheet of players from the 1989 season.  This is to help you choose players that had not retired after the 1988 season: 1989 NFL Players
  • Here is a link to the 1989 draft: 1989 NFL Draft
  • The 8 teams will be divided into two divisions, each with 4 teams.  Each team will play members of their own division twice and members of the other division once.  Ties in regular season will not be broken.
  • The games will be played by using the data from the 1989 season data analysis. Week 1 NFL stats for Week 1, etc.  Here is a link: NFL Box Scores
  • After the 10 game regular season game, each conference will be seeded 1 – 4.  Tiebreakers for seeding will be head-to-head, conference record, total points and lastly coin flip.
  • There will be supplemental drafts after the first 3 games (first round of divisional games),  after the first 7 games (following cross-divisional games) and then after the first 10 games (end of regular season).
  • The playoffs will be weeks 11 – 13 from the 1989 season. Numbers are the team’s seeding.
    • Week 11
      • Game 1 – Conf A – 1 vs 4
      • Game 2 – Conf A – 2 vs 3
      • Game 3 – Conf B – 1 vs 4
      • Game 4 – Conf B – 2 vs 3
    • Week 12
      • Game 5 – Game 1 Winner vs Game 2 Winner
Game 6 – Game 1 Loser vs Game 2 Loser
Game 7 – Game 3 Winner vs Game 4 Winner
      • Game 8 – Game 2 Loser vs Game 3 Loser
    • Week 13 –
      • Game 9 – Game 5 Winner vs Game 7 Winner – Championship and Second
      • Game 10 – Game 5 Loser vs Game 7 Loser – Third and Fourth
      • Game 11 – Game 6 Winner vs Game 8 Winner – Fifth and Sixth
      • Game 12 – Game 6 Loser vs Game 8 Loser – Seventh and Eighth

The challenges were keeping students from looking ahead as well as some rookies in 1989 went on to have fabulous careers so students were aware of certain players who had Hall of Fame careers.  The supplemental draft were the intriguing events and showed students seeking trends in data to help them make their choices.  In particular, they were looking for week to eek consistency, rather than a big total that might have occurred in a single week.

The Mathematics of Fantasy Sports – The NBA Task

This past week, we began the course The Mathematics of Fantasy Sports with 19 young men from grades 10 through 12 and two instructors  Sam Gough and Doug Boomer.  We quickly had students draft a team of NBA players since that was an on g0ing game that students could track  Here was the general lesson plan–

The NBA Challenge – Monday, January 5

  • The class will be divided into 8 teams (3 teams of 3 and 5 teams of 2).
  • Each team will research how various Websites (such as ESPN, Yahoo or CBSSPORTS) allocate points for a fantasy NBA Team that consists of a point guard, shooting guard, small forward, power forward, center and a sixth man that can be any position.
  • Based upon the research, the class will determine how the points will be allocated for the play of 6 NBA positions:  point guard, shooting guard, small forward, power forward, center and a sixth man that can be any position.
  • Once points are decided, an Excel spreadsheet will be created by each team to track their team’s progress.  A partner team will be chosen who will verify your team’s results.
  • A random drawing will occur for the draft order.  The teams will then research players and discuss draft strategy before choosing a team.
  • A serpentine draft will be used for the draft – round 1 will be teams 1 – 8 then round 2 will be teams 8 – 1 and  so forth.  The commissioners (Boomer/Gough) will resolve issues, such as player position, as they arise.
  • Each morning beginning on January 6th through January 23rd, each teams will record their teams results in the Excel spreadsheet from the previous day or days.  Results and daily standings will be announced.
  • At the end of each day, any player that is listed as OUT on their ESPN site can be replaced.  The waiver order will be the reverse of the current overall standings.
  • On Friday, January 9 and January 16, there will be a mini-draft, from worst to first, to replace any player on your roster.
  • The cumulative points from NBA games from January 5th through January 22nd for your team will be used to determine the winner.  A maximum of 60 games played by your team’s 6 man roster will be allowed.

The students showed during the January 9 Friday draft different philosophies about how to use data that was absent during the initial draft.  Each team seams to be using data in different ways to make choices but the key is that they are using data, not just basing choices on reputation or “eye candy”.  A deeper appreciation of data and its usefulness in making decisions is a Learning Outcome for our team.

Constructing Viable Arguments

In a recent article in Education Post, Linda Gojak, who is a 40 year classroom teacher, mathematics professor, and national education leader, stated:

The standards… do describe what it takes to be a good problem solver and to think mathematically.  To solve problems is one of the core reasons we do mathematics.

And in an USA Today column, Solomon Friedberg, the chair of the Math Department at Boston College, states:

Here is what good math learning produces: Students who can compute correctly and wisely, choosing the best way to do a given computation; students who can explain what they are doing when they solve a problem or use math to analyze a situation; and students who have the flexibility and understanding to find the best approach to a new problem.

Both of these statements were defending Common Core and what those standards produce.  I do not adhere to the Common Core Standards, but I clearly strive to help students achieve the big ideas mentioned in the two articles.  I often tell me students that they won’t remember all the results in 10 years but it is hoped that they will benefit form the process of learning mathematics to solve problems.  My classroom continues to be one in which student answers are followed by  teacher questions, such as I wonder why…?  What if….?, that challenge student thinking and attempt to move them from memorizing to seeing the larger picture and the connectedness of mathematics.  This structure leads to the  CCSS Mathematical Practice 3, which is Construct viable arguments and critique the reasoning of others.  I have students who often believe The Answer is all they need.  And some are actually able to get The Answer in numerous ways.  But the ability to make their arguments viable and visible is the step that I want my students to take.  And I hope they can learn how to effectively give feedback to others who are willing to make there work public for all to see.  The learning of mathematics is ever changing and I think changing in a way that will produce better problem-solvers and thinkers.

Making sense of problems and persevering in solving them

The first Common Core Standard for Mathematical Practices is I can make sense of problems and persevere in solving them.  I want all of my students to be able to say this when done with my class.  But how do they know where they are in accomplishing that goal?  From Gough and Wilson, here are the four levels of that progression, with Level 3 being the target:


So, how will that impact my classroom?  I want to be able to give students a problem and they will work collaboratively to make an attempt to a solution (Level 1).  Just the willingness to attempt, knowing they may struggle and possibly fail is a huge step for many students.  Once they have attempted, they may need to ask questions, get clarifications and possibly seek help for others (including me).  But I expect them to keep working and to not give up (Level 2).  Through their continued effort and the willingness to try again, students will begin to make sense of the problems and a solution will become visible to them (Level 3). Once they see the solution one way, they will then strive to solve the problem additional ways (Level 4).  What a great classroom atmosphere when this occurs!  I continually seek problems and settings where this mathematical practice can occur.  I see lots of connection to the Westminster’s Learning for Life  Vision statement.  Students are communicating and collaborating, problem-finding and problem-solving as well as reflecting and revising.  It also reflects the Mathematics Department Mission Statement, which is:

             To cultivate unselfconscious critical thinkers who pursue solutions to unfamiliar problems with persistence and articulate ideas using precise mathematical language.

I believe that my students will benefit greatly by using these levels to reflect on their own learning progression towards the target of “I can make sense of problems and persevere in solving them”.

What Messages do Students Hear?

As I watch Dr. Boaler’s videos from the How to Learn Math online course from Stanford, I realize more and more the effect that messaging has on student beliefs about their own mathematical abilities.  Anything from the toy department in a local store, to the statements made by well-meaning peers, teachers and parents, to course names and even to TV shows or movies.  Do we want students to believe that it is okay not to do well in mathematics?  That mathematics is only for the chosen few? Dr. Boaler believes all students can succeed in math if we help students to develop and embrace a growth mindset.  I embrace that belief as well.  Dr. Boaler often refers to the work of Carol Dweck and her book, Mindset: The New Psychology of Success, which is a great read for anyone.  Dr. Boaler makes Dweck’s work relevant for the learning of mathematics.  Dr. Boaler has also has listed key advice for parents to help students at home on her website,  Here is a quick summary of the advice:

  1. Encourage children to play maths puzzles and games.
  2. Always be encouraging and never tell kids they are wrong when they are working on maths problems. 
  3. Never associate maths with speed. 
  4. Never share with your children the idea that you were bad at maths at school or you dislike it, especially if you are a mother. 
  5. Encourage number sense.
  6. Perhaps most important of all, encourage a growth mindset, ie the idea that ability and smartness change as you work more and learn more.  

Please visit her website for more information about each of the advice steps mentioned.  What students hear greatly effect their own beliefs about what they can and cannot do.  I hope to remain positive and send the message to my students on a daily basis that they can do mathematics.

And a New Year Begins!

A letter to my students’ parents:

I am excited to be your child’s teacher this year in mathematics and have immensely enjoyed the beginning of the year. I take each summer as an opportunity to reflect on my own teaching and how I can best help your child reach his or hers full potential. I enrolled in online at Stanford, in which student thinking about mathematics is explored. Each of my students is taken the student version of the course and I encourage each of you to enroll and join us at How to Learn Math: For Students. In one of the videos, she mentions the Mathematician’s Lament and I very much believe to be an important read as I want my class to be about the artistry of mathematics.

As the year has begun, I want students in my classes to feel comfortable with each other and myself in a collaborative environment. I want them to be willing to try and fail and trust that either a peer or myself will help then in a positive way to grasp with what they maybe struggling. We want to delve deeply into fewer topics with outcome being a deeper, more meaningful understanding.

Your child has many resources at Westminster to help them when they are struggling. I keep the course materials on Schoology. You do not have direct access to the site, but I would encourage you to have your child share the site with you. I am available for office hours on Monday and Friday from 8:00-8:30 and Monday through Thursday from 3:00-3:30. We also have a Math Lab and below is the notice placed in the announcements

“The Math Lab is in Campbell Hall Room 303. It is open for students to drop in for extra help on Mondays, Tuesdays, Thursdays, and Fridays 8:30 – 12:30. In addition, the Mu Alpha Theta Math Club members are available to offer peer tutoring in math and science in the Math Lab on Tuesdays and Thursdays 7:45- 8:20am and Wednesdays 8:10-8:50am.”